Changes between Version 14 and Version 15 of Normalization

Timestamp:

02/04/26 00:23:36 (5 days ago)

Author:

233144

Comment:

--

Normalization

@@ -1 +1 @@
 = Normalization
 
-== Initial de-normalized relation and functional dependencies
-
-Attributes with identical names originating from different relations were renamed using entity-based prefixes in order to avoid duplicate attribute names in the universal relation.
-
-R(user_id, username, password, email,
+== 1. Definition of the Universal Relation
+
+We start from a single universal relation containing all attributes from the domain.
+
+Universal_Relation(
+user_id, username, password, email,
 admin_user_id,
 component_id, component_name, component_brand, component_price, component_type, component_img_url,
@@ -15 +16 @@
 motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots,
 pc_case_cooler_max_height, pc_case_gpu_max_length,
-pc_case_storage_form_factor, case_storage_num_slots,
+pc_case_storage_form_factor, pc_case_storage_num_slots,
 pc_case_ps_form_factor,
 pc_case_mobo_form_factor,
@@ -30 +31 @@
 rating_build_value,
 review_id, review_build_id, review_user_id, review_content, review_created_at,
-suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
-
-Before anything else, we need to note that pc_case_storage_form_factor (composite with case_storage_num_slots), pc_case_ps_form_factor, pc_case_mobo_form_factor, and cooler_cpu_socket are multivalued attributes.
-
-Now we will list the functional dependencies (the primary/composite keys are on the left side). For component_id the name of the proper subclass will be shown in parenthesis for better presentation.
-
-FD1: user_id -> username, password, email
-
-FD2: admin_user_id -> user_id (FK relationship)
-
-FD3: component_id -> component_name, component_brand, component_price, component_type, component_img_url
-
-FD4: component_id(cpu) -> cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp
-
-FD5: component_id(gpu) -> gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length
-
-FD6: component_id(memory) -> memory_type, memory_speed, memory_capacity, memory_modules
-
-FD7: component_id(storage) -> storage_type, storage_capacity, storage_form_factor
-
-FD8: component_id(power_supply) -> power_supply_type, power_supply_wattage, power_supply_form_factor
-
-FD9: component_id(motherboard) -> motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots
-
-FD10: component_id(pc_case) -> pc_case_cooler_max_height, pc_case_gpu_max_length
-
-FD11: component_id(cooler) -> cooler_type, cooler_height, cooler_max_tdp_supported
-
-FD12: component_id(memory_card) -> memory_card_num_slots, memory_card_interface
-
-FD13: component_id(optical_drive) -> optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed
-
-FD14: component_id(sound_card) -> sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel
-
-FD15: component_id(cables) -> cables_length_cm, cables_type
-
-FD16: component_id(network_adapter) -> network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas
-
-FD17: component_id(network_card) -> network_card_num_ports, network_card_speed, network_card_interface
-
-FD18: build_id -> build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
-
-FD19: build_id, component_id -> build_component_num_components
-
-FD20: build_id, user_id -> rating_build_value
-
-FD21: build_id, user_id -> (represents favorite_build relationship, no additional attributes)
-
-FD22: review_id -> review_build_id, review_user_id, review_content, review_created_at
-
-FD23: review_build_id, review_user_id -> review_id (alternate key constraint from UNIQUE)
-
-FD24: suggestion_id -> suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
-
-Left: user_id, admin_user_id, component_id, build_id, review_id, suggestion_id
-
-Left and right: build_user_id, user_id (via various FDs), review_build_id, review_user_id
-
-Right:
-username, password, email, component_name, component_brand, component_price, component_type, component_img_url, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length, memory_type, memory_speed, memory_capacity, memory_modules, storage_type, storage_capacity, storage_form_factor, power_supply_type, power_supply_wattage, power_supply_form_factor, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots, pc_case_cooler_max_height, pc_case_gpu_max_length, case_storage_num_slots, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_type, cooler_height, cooler_max_tdp_supported, cooler_cpu_socket, memory_card_num_slots, memory_card_interface, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel, cables_length_cm, cables_type, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas, network_card_num_ports, network_card_speed, network_card_interface, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved, build_component_num_components, rating_build_value, review_build_id, review_user_id, review_content, review_created_at, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
-
-Attributes that never appear on the right side (only left) are user_id, admin_user_id, component_id, build_id, review_id, suggestion_id. This means that they must be in every candidate key because nothing can determine them.
-
-Let's compute the closures of each individual left-side attribute to see what we can derive from each:
-
-{user_id}+ : By applying FD1 we add username, password, email. So {user_id}+ = {user_id, username, password, email}. This is NOT a superkey (missing all other attributes).
-
-{admin_user_id}+ : By applying FD2, admin_user_id → user_id. By applying FD1, user_id → username, password, email. So {admin_user_id}+ = {admin_user_id, user_id, username, password, email}. This is NOT a superkey (missing component_id, build_id, review_id, suggestion_id and their dependents).
-
-{component_id}+ : By applying FD3-FD17 we add component_name, component_brand, component_price, component_type, component_img_url, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length, memory_type, memory_speed, memory_capacity, memory_modules, storage_type, storage_capacity, storage_form_factor, power_supply_type, power_supply_wattage, power_supply_form_factor, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots, pc_case_cooler_max_height, pc_case_gpu_max_length, cooler_type, cooler_height, cooler_max_tdp_supported, memory_card_num_slots, memory_card_interface, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel, cables_length_cm, cables_type, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas, network_card_num_ports, network_card_speed, network_card_interface. So {component_id}+ contains all component attributes but is NOT a superkey (missing user_id, admin_user_id, build_id, review_id, suggestion_id and their dependents).
-
-{build_id}+ : By applying FD18 we add build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved. By applying FD19 we need component_id which we don't have. By applying FD20 and FD21 we need user_id which we don't have. So {build_id}+ = {build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved}. This is NOT a superkey.
-
-{review_id}+ : By applying FD22 we add review_build_id, review_user_id, review_content, review_created_at. We cannot derive build_id or user_id directly from the relation. So {review_id}+ = {review_id, review_build_id, review_user_id, review_content, review_created_at}. This is NOT a superkey.
-
-{suggestion_id}+ : By applying FD24 we add suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type. We cannot derive admin_user_id directly. So {suggestion_id}+ = {suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type}. This is NOT a superkey.
-
-Since no single left-side attribute can determine all others, we must combine them. Let's test the candidate key K = {user_id, admin_user_id, component_id, review_id, suggestion_id, build_id}.
-
-We now compute the closure K+
-
-Start with K+ = {user_id, admin_user_id, component_id, review_id, suggestion_id, build_id}
-
-By applying FD1 – we add:
-username, password, email
-
-By applying FD2 – we verify admin specialization (admin_user_id is FK to user_id, already in set)
-
-By applying FD3 – FD17, we add:
-component_name, component_brand, component_price, component_type, component_img_url,
-cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp,
-gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length,
-memory_type, memory_speed, memory_capacity, memory_modules,
-storage_type, storage_capacity, storage_form_factor,
-power_supply_type, power_supply_wattage, power_supply_form_factor,
-motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots,
-pc_case_cooler_max_height, pc_case_gpu_max_length,
-cooler_type, cooler_height, cooler_max_tdp_supported,
-memory_card_num_slots, memory_card_interface,
-optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed,
-sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel,
-cables_length_cm, cables_type,
-network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas,
-network_card_num_ports, network_card_speed, network_card_interface
-
-By applying FD18, we add:
-build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
-
-By applying FD19 (we have build_id and component_id), we add:
-build_component_num_components
-
-By applying FD20 and FD21 (we have build_id and user_id), we add:
-rating_build_value
-
-By applying FD22, we add:
-review_build_id, review_user_id, review_content, review_created_at
-
-By applying FD24, we add:
-suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
-
-After excluding multivalued attributes which we handle separately in 1NF, K+ contains all attributes of R, which means that K is a superkey.
-
-To verify that K = {user_id, admin_user_id, component_id, review_id, suggestion_id, build_id} is a candidate key, we test all proper subsets:
-
-{admin_user_id, component_id, review_id, suggestion_id, build_id}+ is missing user_id which blocks FD20 and FD21 (need user_id for build_id, user_id -> rating_build_value and favorite_build). 
-
-{user_id, component_id, review_id, suggestion_id, build_id}+ is missing admin_user_id which blocks FD24 completely (need to derive suggestion_admin_id from admin_user_id). 
-
-{user_id, admin_user_id, review_id, suggestion_id, build_id}+ is missing component_id which blocks FD19 (need build_id, component_id for build_component_num_components) and all component attributes. 
-
-{user_id, admin_user_id, component_id, suggestion_id, build_id}+ is missing review_id which blocks FD22 (need review_id -> review attributes). 
-
-{user_id, admin_user_id, component_id, review_id, build_id}+ is missing suggestion_id which blocks FD24 (need suggestion_id -> suggestion attributes). 
-
-{user_id, admin_user_id, component_id, review_id, suggestion_id}+ is missing build_id which blocks FD18, FD19, FD20, FD21 (need build_id -> build attributes and build_id for various composite FDs). 
-
-Therefore, K = {user_id, admin_user_id, component_id, review_id, suggestion_id, build_id} is the minimal candidate key (no proper subset is a superkey).
-
-The universal relation violates 1NF because the attributes pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, and cooler_cpu_socket represent multivalued attributes, and can contain multiple values for a single component_id. This will be resolved by decomposing them into separate relations.
-
-== 1NF Decomposition
-
-To achieve 1NF, we decompose the multivalued attributes from R into:
-
-component_id, pc_case_storage_form_factor -> case_storage_num_slots
-
-component_id, pc_case_ps_form_factor -> no non-key attributes
-
-component_id, pc_case_mobo_form_factor -> no non-key attributes
-
-component_id, cooler_cpu_socket -> no non-key attributes
-
-1NF is satisfied now. 2NF is not satisfied because of partial dependencies. We can see this through the functional dependencies:
-
-FD1: user_id is part of the key, but these attributes such as username, password, email depend only on user_id.
-
-FD2: admin_user_id is part of the key, but user_id depends only on admin_user_id.
-
-FD3-FD17: component_id is part of the key, but these attributes such as component_name, component_brand, component_price etc. depend only on component_id.
-
-FD18: build_id is part of the key, but these attributes such as build_user_id, build_name etc. depend only on build_id.
-
-FD19: (build_id, component_id) is part of the key, but build_component_num_components depends on (build_id, component_id).
-
-FD20: (build_id, user_id) is part of the key, but rating_build_value depends on (build_id, user_id).
-
-FD21: (build_id, user_id) is part of the key, but represents favorite_build (no attributes).
-
-FD22: review_id is part of the key, but these attributes such as review_build_id, review_user_id, review_content, review_created_at depend only on review_id.
-
-FD24: suggestion_id is part of the key, but these attributes such as suggestion_user_id, suggestion_admin_id etc. depend only on suggestion_id.
-
-We will decompose by creating new relations from R following FD1, FD2 to FD24 where the attributes will depend only on the determinant attribute(s), which will become the primary key:
-
-user_id -> username, password, email
-
-admin_user_id -> user_id
-
-component_id -> component_name, component_brand, component_price, component_type, component_img_url
-
-component_id -> cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp
-
-component_id -> gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length
-
-component_id -> memory_type, memory_speed, memory_capacity, memory_modules
-
-component_id -> storage_type, storage_capacity, storage_form_factor
-
-component_id -> power_supply_type, power_supply_wattage, power_supply_form_factor
-
-component_id -> motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots
-
-component_id -> pc_case_cooler_max_height, pc_case_gpu_max_length
-
-build_id -> build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
-
-build_id, component_id -> build_component_num_components
-
-build_id, user_id -> rating_build_value
-
-build_id, user_id -> (represents favorite_build)
-
-review_id -> review_build_id, review_user_id, review_content, review_created_at
-
-suggestion_id -> suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
-
-Primary key in all of these is the attribute(s) on the left side.
-
-== 2NF Decomposition
-
-All these partial dependencies are addressed in 2NF. We will solve these partial dependencies by creating new relations where each determinant becomes a primary key:
-
-R1 { user_id, username, password, email } where PK = user_id - BCNF
-
-R2 { admin_user_id, user_id } where PK = admin_user_id, FK = user_id -> R1.user_id - BCNF
-
-R3 { component_id, component_name, component_brand, component_price, component_type, component_img_url } where PK = component_id - BCNF
-
-R4 { component_id, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp } where PK = component_id - BCNF
-
-R5 { component_id, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length } where PK = component_id - BCNF
-
-R6 { component_id, memory_type, memory_speed, memory_capacity, memory_modules } where PK = component_id - BCNF
-
-R7 { component_id, storage_type, storage_capacity, storage_form_factor } where PK = component_id - BCNF
-
-R8 { component_id, power_supply_type, power_supply_wattage, power_supply_form_factor } where PK = component_id - BCNF
-
-R9 { component_id, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots } where PK = component_id - BCNF
-
-R10 { component_id, pc_case_cooler_max_height, pc_case_gpu_max_length } where PK = component_id - BCNF
-
-R11 { build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved } where PK = build_id - BCNF
-
-R12 { build_id, component_id, build_component_num_components } where PK = (build_id, component_id) - BCNF
-
-R13 { build_id, user_id, rating_build_value } where PK = (build_id, user_id) - BCNF
-
-R14 { build_id, user_id } where PK = (build_id, user_id) - BCNF
-
-R15 { component_id, cooler_type, cooler_height, cooler_max_tdp_supported } where PK = component_id - BCNF
-
-R16 { component_id, memory_card_num_slots, memory_card_interface } where PK = component_id - BCNF
-
-R17 { component_id, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed } where PK = component_id - BCNF
-
-R18 { component_id, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel } where PK = component_id - BCNF
-
-R19 { component_id, cables_length_cm, cables_type } where PK = component_id - BCNF
-
-R20 { component_id, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas } where PK = component_id - BCNF
-
-R21 { component_id, network_card_num_ports, network_card_speed, network_card_interface } where PK = component_id - BCNF
-
-R22 { review_id, review_build_id, review_user_id, review_content, review_created_at } where PK = review_id, Alt Key = (review_build_id, review_user_id) - BCNF
-
-R23 { suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type } where PK = suggestion_id - BCNF
-
-The remainder of R { component_id, review_id, suggestion_id, build_id, user_id, admin_user_id } where PK = K (component_id, review_id, suggestion_id, build_id, user_id, admin_user_id)
-
-2NF is satisfied for all relations. All partial dependencies have been eliminated. No transitive dependencies can exist in relations with only key attributes or attributes derived entirely from the key.
-
-All relations are already in BCNF since every determinant of every functional dependency is a candidate key of its respective relation.
-
-== Final BCNF relations
-
-Users { user_id, username, password, email }
-
-Admins { admin_user_id, user_id (FK to Users.user_id) }
-
-Components { component_id, component_name, component_brand, component_price, component_type, component_img_url }
-
-Cpu { component_id, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp }
-
-Gpu { component_id, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length }
-
-Memory { component_id, memory_type, memory_speed, memory_capacity, memory_modules }
-
-Storage { component_id, storage_type, storage_capacity, storage_form_factor }
-
-Power_Supply { component_id, power_supply_type, power_supply_wattage, power_supply_form_factor }
-
-Motherboard { component_id, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots }
-
-Pc_Case { component_id, pc_case_cooler_max_height, pc_case_gpu_max_length }
-
-Case_Storage_Form_Factors { component_id, pc_case_storage_form_factor, case_storage_num_slots }
-
-Case_Ps_Form_Factors { component_id, pc_case_ps_form_factor }
-
-Case_Mobo_Form_Factors { component_id, pc_case_mobo_form_factor }
-
-Cooler { component_id, cooler_type, cooler_height, cooler_max_tdp_supported }
-
-Cooler_Cpu_Sockets { component_id, cooler_cpu_socket }
-
-Memory_Card { component_id, memory_card_num_slots, memory_card_interface }
-
-Optical_Drive { component_id, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed }
-
-Sound_Card { component_id, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel }
-
-Cables { component_id, cables_length_cm, cables_type }
-
-Network_Adapter { component_id, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas }
-
-Network_Card { component_id, network_card_num_ports, network_card_speed, network_card_interface }
-
-Build { build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved }
-
-Build_Component { build_id, component_id, build_component_num_components }
-
-Rating_Build { build_id, user_id, rating_build_value }
-
-Favorite_Build { build_id, user_id }
-
-Review { review_id, review_build_id, review_user_id, review_content, review_created_at }
-
-Suggestions { suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type }
-
+suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+)
+
+== 2. Functional Dependencies
+
+We identify all functional dependencies that hold in the universal relation:
+
+F = {
+
+FD1: user_id → username, password, email
+
+FD2: admin_user_id → ∅
+
+FD3: component_id → component_name, component_brand, component_price, component_type, component_img_url
+
+FD4: component_id → cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp
+
+FD5: component_id → gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length
+
+FD6: component_id → memory_type, memory_speed, memory_capacity, memory_modules
+
+FD7: component_id → storage_type, storage_capacity, storage_form_factor
+
+FD8: component_id → power_supply_type, power_supply_wattage, power_supply_form_factor
+
+FD9: component_id → motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots
+
+FD10: component_id → pc_case_cooler_max_height, pc_case_gpu_max_length
+
+FD11: component_id → cooler_type, cooler_height, cooler_max_tdp_supported
+
+FD12: component_id → memory_card_num_slots, memory_card_interface
+
+FD13: component_id → optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed
+
+FD14: component_id → sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel
+
+FD15: component_id → cables_length_cm, cables_type
+
+FD16: component_id → network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas
+
+FD17: component_id → network_card_num_ports, network_card_speed, network_card_interface
+
+FD18: build_id → build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
+
+FD19: (build_id, component_id) → build_component_num_components
+
+FD20: (build_id, user_id) → rating_build_value
+
+FD21: (build_id, user_id) → ∅ (represents favorite_build relationship)
+
+FD22: review_id → review_build_id, review_user_id, review_content, review_created_at
+
+FD23: suggestion_id → suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+
+FD24: (component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots
+
+FD25: (component_id, pc_case_mobo_form_factor) → ∅
+
+FD26: (component_id, pc_case_ps_form_factor) → ∅
+
+FD27: (component_id, cooler_cpu_socket) → ∅
+
+}
+
+== 3. Candidate Keys and Primary Key
+
+=== 3.1 Determination of Candidate Keys
+
+To find a candidate key, we must find a minimal set of attributes whose closure contains all attributes of the relation.
+
+=== 3.2 Attribute Classification (Left / Right Side)
+
+|| Attribute =||= Left Side =||= Right Side =||= Classification =||
+|| user_id || ✓ (FD1) || ✗ || Left only ||
+|| username || ✗ || ✓ (FD1) || Right only ||
+|| password || ✗ || ✓ (FD1) || Right only ||
+|| email || ✗ || ✓ (FD1) || Right only ||
+|| admin_user_id || ✓ (FD2) || ✗ || Left only ||
+|| component_id || ✓ (FD3-FD17, FD24-FD27) || ✗ || Left only ||
+|| component_name || ✗ || ✓ (FD3) || Right only ||
+|| component_brand || ✗ || ✓ (FD3) || Right only ||
+|| component_price || ✗ || ✓ (FD3) || Right only ||
+|| component_type || ✗ || ✓ (FD3) || Right only ||
+|| component_img_url || ✗ || ✓ (FD3) || Right only ||
+|| [CPU attributes] || ✗ || ✓ (FD4) || Right only ||
+|| [GPU attributes] || ✗ || ✓ (FD5) || Right only ||
+|| [Memory attributes] || ✗ || ✓ (FD6) || Right only ||
+|| [Storage attributes] || ✗ || ✓ (FD7) || Right only ||
+|| [Power Supply attributes] || ✗ || ✓ (FD8) || Right only ||
+|| [Motherboard attributes] || ✗ || ✓ (FD9) || Right only ||
+|| pc_case_cooler_max_height || ✗ || ✓ (FD10) || Right only ||
+|| pc_case_gpu_max_length || ✗ || ✓ (FD10) || Right only ||
+|| pc_case_storage_form_factor || ✓ (FD24) || ✗ || Left only ||
+|| pc_case_storage_num_slots || ✗ || ✓ (FD24) || Right only ||
+|| pc_case_ps_form_factor || ✓ (FD26) || ✗ || Left only ||
+|| pc_case_mobo_form_factor || ✓ (FD25) || ✗ || Left only ||
+|| cooler_type || ✗ || ✓ (FD11) || Right only ||
+|| cooler_height || ✗ || ✓ (FD11) || Right only ||
+|| cooler_max_tdp_supported || ✗ || ✓ (FD11) || Right only ||
+|| cooler_cpu_socket || ✓ (FD27) || ✗ || Left only ||
+|| [Memory Card attributes] || ✗ || ✓ (FD12) || Right only ||
+|| [Optical Drive attributes] || ✗ || ✓ (FD13) || Right only ||
+|| [Sound Card attributes] || ✗ || ✓ (FD14) || Right only ||
+|| [Cables attributes] || ✗ || ✓ (FD15) || Right only ||
+|| [Network Adapter attributes] || ✗ || ✓ (FD16) || Right only ||
+|| [Network Card attributes] || ✗ || ✓ (FD17) || Right only ||
+|| build_id || ✓ (FD18-FD21) || ✗ || Left only ||
+|| build_user_id || ✗ || ✓ (FD18) || Right only ||
+|| build_name || ✗ || ✓ (FD18) || Right only ||
+|| build_created_at || ✗ || ✓ (FD18) || Right only ||
+|| build_description || ✗ || ✓ (FD18) || Right only ||
+|| build_total_price || ✗ || ✓ (FD18) || Right only ||
+|| build_is_approved || ✗ || ✓ (FD18) || Right only ||
+|| build_component_num_components || ✗ || ✓ (FD19) || Right only ||
+|| rating_build_value || ✗ || ✓ (FD20) || Right only ||
+|| review_id || ✓ (FD22) || ✗ || Left only ||
+|| review_build_id || ✗ || ✓ (FD22) || Right only ||
+|| review_user_id || ✗ || ✓ (FD22) || Right only ||
+|| review_content || ✗ || ✓ (FD22) || Right only ||
+|| review_created_at || ✗ || ✓ (FD22) || Right only ||
+|| suggestion_id || ✓ (FD23) || ✗ || Left only ||
+|| suggestion_user_id || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_admin_id || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_link || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_admin_comment || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_description || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_status || ✗ || ✓ (FD23) || Right only ||
+|| suggestion_component_type || ✗ || ✓ (FD23) || Right only ||
+
+
+=== 3.3 Attributes That Appear ONLY on the Left Side
+
+The following attributes appear only on the left side of functional dependencies and never on the right side:
+
+- user_id – must be part of every candidate key
+- component_id – must be part of every candidate key
+- build_id – must be part of every candidate key
+- review_id – must be part of every candidate key
+- suggestion_id – must be part of every candidate key
+- pc_case_storage_form_factor – must be part of every candidate key
+- pc_case_ps_form_factor – must be part of every candidate key
+- pc_case_mobo_form_factor – must be part of every candidate key
+- cooler_cpu_socket – must be part of every candidate key
+- admin_user_id – must be part of every candidate key
+
+=== 3.4 Closure Computation
+
+**Step 1:** We start with {user_id, component_id, build_id, review_id, suggestion_id, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket, admin_user_id} and compute the closure:
+
+{user_id, component_id, build_id, review_id, suggestion_id, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket, admin_user_id}⁺:
+
+- From FD1 (user_id → username, password, email): We obtain username, password, email
+- From FD2 (admin_user_id → ∅): No new attributes
+- From FD3–FD17 (component_id → ...): We obtain component_name, component_brand, component_price, component_type, component_img_url, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length, memory_type, memory_speed, memory_capacity, memory_modules, storage_type, storage_capacity, storage_form_factor, power_supply_type, power_supply_wattage, power_supply_form_factor, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots, pc_case_cooler_max_height, pc_case_gpu_max_length, cooler_type, cooler_height, cooler_max_tdp_supported, memory_card_num_slots, memory_card_interface, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel, cables_length_cm, cables_type, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas, network_card_num_ports, network_card_speed, network_card_interface
+- From FD18 (build_id → ...): We obtain build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
+- From FD19 ((build_id, component_id) → build_component_num_components): We obtain build_component_num_components (both build_id and component_id already in closure)
+- From FD20 ((build_id, user_id) → rating_build_value): We obtain rating_build_value (both build_id and user_id already in closure)
+- From FD21 ((build_id, user_id) → ∅): No new attributes
+- From FD22 (review_id → ...): We obtain review_build_id, review_user_id, review_content, review_created_at
+- From FD23 (suggestion_id → ...): We obtain suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+- From FD24 ((component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots): We obtain pc_case_storage_num_slots (both attributes already in closure)
+- From FD25–FD27 (composite FDs with empty RHS): No new attributes
+
+Closure = Universal_Relation ✓ (all attributes are present)
+
+=== 3.5 Minimality Check
+
+Testing proper subsets of K = {user_id, component_id, build_id, review_id, suggestion_id, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket, admin_user_id}:
+
+|| Subset =||= Closure Equals Universal_Relation? =||= Justification =||
+|| K − {user_id} || ✗ NO || Cannot derive username, password, email (FD1 requires user_id) ||
+|| K − {component_id} || ✗ NO || Cannot derive component attributes (FD3–FD17 require component_id) ||
+|| K − {build_id} || ✗ NO || Cannot derive build attributes (FD18 requires build_id) ||
+|| K − {review_id} || ✗ NO || Cannot derive review attributes (FD22 requires review_id) ||
+|| K − {suggestion_id} || ✗ NO || Cannot derive suggestion attributes (FD23 requires suggestion_id) ||
+|| K − {pc_case_storage_form_factor} || ✗ NO || FD24 requires pc_case_storage_form_factor ||
+|| K − {pc_case_ps_form_factor} || ✗ NO || FD26 requires pc_case_ps_form_factor ||
+|| K − {pc_case_mobo_form_factor} || ✗ NO || FD25 requires pc_case_mobo_form_factor ||
+|| K − {cooler_cpu_socket} || ✗ NO || FD27 requires cooler_cpu_socket ||
+|| K − {admin_user_id} || ✗ NO || admin_user_id is a left-only attribute and must be in every candidate key ||
+
+
+**Conclusion:** K = {user_id, component_id, build_id, review_id, suggestion_id, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket, admin_user_id} is minimal and is the only candidate key.
+
+=== 3.6 Choice of Primary Key
+
+**Chosen Primary Key:** {user_id, component_id, build_id, review_id, suggestion_id, pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket, admin_user_id}
+
+**Justification:** This is the primary key (minimal candidate key) because only with these attributes can we derive all remaining attributes from the universal relation through the closure of functional dependencies.
+
+---
+
+== 4. 1NF Decomposition
+
+=== 4.1 1NF Violation Identification
+
+The relation violates 1NF because the attributes pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, and cooler_cpu_socket represent multivalued attributes. Each component_id can have multiple values for each of these attributes.
+
+=== 4.2 1NF Decomposition
+
+==== Relation R1: Case_Storage_Form_Factors
+
+Case_Storage_Form_Factors(component_id, pc_case_storage_form_factor, pc_case_storage_num_slots)
+PK: (component_id, pc_case_storage_form_factor)
+
+**Attributes Included:** component_id, pc_case_storage_form_factor, pc_case_storage_num_slots
+
+**Functional Dependency Enforced:** FD24: (component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots
+
+**Lossless Join Check:**
+- This relation is decomposed directly from the universal relation by extracting the multivalued attribute group pc_case_storage_form_factor
+- The composite primary key (component_id, pc_case_storage_form_factor) uniquely identifies each tuple
+- When joined back with the main relation via component_id, no spurious tuples are introduced
+- The decomposition captures the exact relationship defined by FD24
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD24: (component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots
+  - Determinant: (component_id, pc_case_storage_form_factor)
+  - Dependent: pc_case_storage_num_slots
+  - All attributes of FD24 appear together in this relation ✓
+- The functional dependency is directly enforceable within this schema
+- **Result: PRESERVED ✓**
+
+---
+
+==== Relation R2: Case_Ps_Form_Factors
+
+Case_Ps_Form_Factors(component_id, pc_case_ps_form_factor)
+PK: (component_id, pc_case_ps_form_factor)
+
+**Attributes Included:** component_id, pc_case_ps_form_factor
+
+**Functional Dependency Enforced:** FD26: (component_id, pc_case_ps_form_factor) → ∅
+
+**Lossless Join Check:**
+- Decomposed directly from the multivalued attribute pc_case_ps_form_factor
+- The primary key (component_id, pc_case_ps_form_factor) uniquely identifies the relationship
+- Joining with the main relation on component_id preserves all original tuples
+- No information loss occurs during decomposition or reconstruction
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD26: (component_id, pc_case_ps_form_factor) → ∅
+  - This is a relationship-only functional dependency with empty RHS
+  - The determinant (component_id, pc_case_ps_form_factor) is present as the primary key
+  - All attributes involved in FD26 are contained in this relation ✓
+- **Result: PRESERVED ✓**
+
+---
+
+==== Relation R3: Case_Mobo_Form_Factors
+
+Case_Mobo_Form_Factors(component_id, pc_case_mobo_form_factor)
+PK: (component_id, pc_case_mobo_form_factor)
+
+**Attributes Included:** component_id, pc_case_mobo_form_factor
+
+**Functional Dependency Enforced:** FD25: (component_id, pc_case_mobo_form_factor) → ∅
+
+**Lossless Join Check:**
+- Decomposed from the multivalued attribute pc_case_mobo_form_factor
+- The composite key fully represents the relationship between component and motherboard form factor
+- Natural join on component_id with the main relation produces no spurious tuples
+- Complete reconstruction of the original relationship is possible
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD25: (component_id, pc_case_mobo_form_factor) → ∅
+  - The relationship-only FD is preserved through the composite primary key
+  - All attributes of FD25 present in this schema ✓
+- **Result: PRESERVED ✓**
+
+---
+
+==== Relation R4: Cooler_Cpu_Sockets
+
+Cooler_Cpu_Sockets(component_id, cooler_cpu_socket)
+PK: (component_id, cooler_cpu_socket)
+
+**Attributes Included:** component_id, cooler_cpu_socket
+
+**Functional Dependency Enforced:** FD27: (component_id, cooler_cpu_socket) → ∅
+
+**Lossless Join Check:**
+- Decomposed directly from the multivalued attribute cooler_cpu_socket
+- The primary key captures the complete relationship
+- Rejoining via component_id preserves all original tuples without duplication
+- No information loss in decomposition or reconstruction
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD27: (component_id, cooler_cpu_socket) → ∅
+  - The relationship-only FD is preserved by the composite key structure
+  - All attributes involved in FD27 are present ✓
+- **Result: PRESERVED ✓**
+
+---
+
+==== Relation R5: Universal_Relation_1NF
+
+Universal_Relation_1NF(
+  user_id, username, password, email,
+  admin_user_id,
+  component_id, component_name, component_brand, component_price, component_type, component_img_url,
+  cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp,
+  gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length,
+  memory_type, memory_speed, memory_capacity, memory_modules,
+  storage_type, storage_capacity, storage_form_factor,
+  power_supply_type, power_supply_wattage, power_supply_form_factor,
+  motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots,
+  pc_case_cooler_max_height, pc_case_gpu_max_length,
+  pc_case_storage_num_slots,
+  cooler_type, cooler_height, cooler_max_tdp_supported,
+  memory_card_num_slots, memory_card_interface,
+  optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed,
+  sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel,
+  cables_length_cm, cables_type,
+  network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas,
+  network_card_num_ports, network_card_speed, network_card_interface,
+  build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved,
+  build_component_num_components,
+  rating_build_value,
+  review_id, review_build_id, review_user_id, review_content, review_created_at,
+  suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+)
+PK: (user_id, component_id, build_id, review_id, suggestion_id, admin_user_id)
+
+**Attributes Included:** All original attributes except the multivalued ones (pc_case_storage_form_factor, pc_case_ps_form_factor, pc_case_mobo_form_factor, cooler_cpu_socket)
+
+**Functional Dependencies Enforced:** FD1–FD24 (all except FD25–FD27)
+
+**Lossless Join Check:**
+- The four decomposed multivalued relations can be joined back to this relation via component_id
+- Each tuple in this main relation corresponds to a set of tuples in each multivalued relation
+- The natural join R5 ⋈ R1 ⋈ R2 ⋈ R3 ⋈ R4 reconstructs the original universal relation without spurious tuples
+- All join paths are on component_id, a common attribute present in all relations
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD1: user_id → username, password, email — All attributes present ✓
+- FD2: admin_user_id → ∅ — Determinant present ✓
+- FD3–FD23: component_id or composite keys → dependent attributes — All attributes present ✓
+- FD24: (component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots — pc_case_storage_num_slots retained ✓
+- All functional dependencies relevant to this relation are preserved
+- **Result: PRESERVED ✓**
+
+---
+
+== 5. 2NF Decomposition
+
+=== 5.1 2NF Violation Analysis
+
+The Universal_Relation_1NF (R5) violates 2NF due to partial dependencies.
+
+**Primary Key:** (user_id, component_id, build_id, review_id, suggestion_id, admin_user_id)
+
+A partial dependency exists when a non-key attribute depends on only a **proper subset** of the composite primary key rather than the entire key.
+
+**Identified Partial Dependencies:**
+
+|| FD =||= Determinant =||= Proper Subset? =||= Non-key Attributes Dependent =||= 2NF Violation? =||
+|| FD1 || user_id || ✓ YES || username, password, email || ✓ YES ||
+|| FD2 || admin_user_id || ✓ YES || (specialization) || ✓ YES ||
+|| FD3–FD17 || component_id || ✓ YES || component attributes || ✓ YES ||
+|| FD18 || build_id || ✓ YES || build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved || ✓ YES ||
+|| FD22 || review_id || ✓ YES || review_build_id, review_user_id, review_content, review_created_at || ✓ YES ||
+|| FD23 || suggestion_id || ✓ YES || suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type || ✓ YES ||
+|| FD19 || (build_id, component_id) || ✓ YES || build_component_num_components || ✓ YES ||
+|| FD20 || (build_id, user_id) || ✓ YES || rating_build_value || ✓ YES ||
+|| FD21 || (build_id, user_id) || ✓ YES || (relationship only) || ✓ YES ||
+
+=== 5.2 2NF Decomposition
+
+We eliminate partial dependencies by creating new relations where each partial dependency determinant becomes a primary key.
+
+---
+
+==== 2NF Relation 1: Users
+
+Users(user_id, username, password, email)
+PK: user_id
+
+**Derived From:** FD1 (user_id → username, password, email)
+
+**Functional Dependency Enforced:** FD1
+
+**Lossless Join Check:**
+- Before decomposition, user_id was part of the composite key in Universal_Relation_1NF
+- Now user_id becomes the sole primary key in Users
+- Intersection of Users and Universal_Relation_1NF: I = {user_id}
+- FD1 guarantees: user_id → username, password, email
+- The join condition is on user_id, which is a candidate key for Users
+- Natural join Users ⋈ Universal_Relation_1NF on user_id produces no spurious tuples
+- All original tuples can be reconstructed: ✓
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD1: user_id → username, password, email
+  - Determinant (user_id) and all dependent attributes (username, password, email) are present in Users ✓
+  - The FD is directly enforceable within this relation
+- No other FD involves only user_id and its dependents in Universal_Relation_1NF
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 2: Admins
+
+Admins(admin_user_id)
+PK: admin_user_id
+FK: admin_user_id → Users.user_id
+
+**Derived From:** FD2 (admin_user_id → ∅)
+
+**Functional Dependency Enforced:** FD2
+
+**Lossless Join Check:**
+- admin_user_id was part of the composite key in Universal_Relation_1NF
+- Now admin_user_id becomes the sole primary key in Admins
+- Intersection: I = {admin_user_id}
+- FD2 is a relationship-only dependency with empty dependent set
+- The join Admins ⋈ Universal_Relation_1NF on admin_user_id reconstructs the relationship
+- No spurious tuples introduced
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD2: admin_user_id → ∅
+  - The determinant (admin_user_id) is the primary key ✓
+  - This relationship-only FD is preserved through the key structure
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 3: Components
+
+Components(component_id, component_name, component_brand, component_price, component_type, component_img_url)
+PK: component_id
+
+**Derived From:** FD3 (component_id → component_name, component_brand, component_price, component_type, component_img_url)
+
+**Functional Dependency Enforced:** FD3
+
+**Lossless Join Check:**
+- component_id was part of the composite key in Universal_Relation_1NF
+- Now component_id becomes the sole primary key in Components
+- Intersection: I = {component_id}
+- FD3 guarantees: component_id uniquely determines all dependent attributes
+- Join Components ⋈ Universal_Relation_1NF on component_id is lossless
+- No spurious tuples introduced
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD3: component_id → component_name, component_brand, component_price, component_type, component_img_url
+  - All attributes of FD3 present in Components ✓
+  - The FD is directly enforceable within this relation
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 4: Cpu
+
+Cpu(component_id, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD4 (component_id → cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp)
+
+**Functional Dependency Enforced:** FD4
+
+**Lossless Join Check:**
+- component_id is a proper subset of the original composite key
+- component_id becomes the primary key in Cpu
+- Intersection: I = {component_id}
+- FD4 ensures all non-key attributes are uniquely determined by component_id
+- Join with Universal_Relation_1NF on component_id is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD4: component_id → cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp
+  - All attributes present in Cpu ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 5: Gpu
+
+Gpu(component_id, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD5
+
+**Lossless Join Check:** LOSSLESS ✓ (same reasoning as Cpu)
+
+**Dependency Preservation Check:** FD5 directly enforceable ✓ PRESERVED ✓
+
+---
+
+==== 2NF Relation 6: Memory
+
+Memory(component_id, memory_type, memory_speed, memory_capacity, memory_modules)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD6
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 7: Storage
+
+Storage(component_id, storage_type, storage_capacity, storage_form_factor)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD7
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 8: Power_Supply
+
+Power_Supply(component_id, power_supply_type, power_supply_wattage, power_supply_form_factor)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD8
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 9: Motherboard
+
+Motherboard(component_id, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD9
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 10: Pc_Case
+
+Pc_Case(component_id, pc_case_cooler_max_height, pc_case_gpu_max_length)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD10
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 11: Cooler
+
+Cooler(component_id, cooler_type, cooler_height, cooler_max_tdp_supported)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD11
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 12: Memory_Card
+
+Memory_Card(component_id, memory_card_num_slots, memory_card_interface)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD12
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 13: Optical_Drive
+
+Optical_Drive(component_id, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD13
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 14: Sound_Card
+
+Sound_Card(component_id, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD14
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 15: Cables
+
+Cables(component_id, cables_length_cm, cables_type)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD15
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 16: Network_Adapter
+
+Network_Adapter(component_id, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD16
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 17: Network_Card
+
+Network_Card(component_id, network_card_num_ports, network_card_speed, network_card_interface)
+PK: component_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD17
+
+**Lossless Join Check:** LOSSLESS ✓
+
+**Dependency Preservation Check:** PRESERVED ✓
+
+---
+
+==== 2NF Relation 18: Build
+
+Build(build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved)
+PK: build_id
+FK: build_user_id → Users.user_id
+
+**Derived From:** FD18 (build_id → build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved)
+
+**Functional Dependency Enforced:** FD18
+
+**Lossless Join Check:**
+- build_id was part of the composite key in Universal_Relation_1NF
+- Now build_id becomes the sole primary key in Build
+- Intersection: I = {build_id}
+- FD18 guarantees all non-key attributes are uniquely determined by build_id
+- Join Build ⋈ Universal_Relation_1NF on build_id is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD18: build_id → build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
+  - All attributes present in Build ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 19: Build_Component
+
+Build_Component(build_id, component_id, build_component_num_components)
+PK: (build_id, component_id)
+FK: build_id → Build.build_id
+FK: component_id → Components.component_id
+
+**Derived From:** FD19 ((build_id, component_id) → build_component_num_components)
+
+**Functional Dependency Enforced:** FD19
+
+**Lossless Join Check:**
+- (build_id, component_id) is NOT a proper subset; it equals the original combination for this FD
+- The composite key becomes the primary key of Build_Component
+- Intersection: I = {build_id, component_id}
+- FD19 guarantees the composite key determines the non-key attribute
+- Join with Universal_Relation_1NF on (build_id, component_id) is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD19: (build_id, component_id) → build_component_num_components
+  - All attributes present ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 20: Rating_Build
+
+Rating_Build(build_id, user_id, rating_build_value)
+PK: (build_id, user_id)
+FK: build_id → Build.build_id
+FK: user_id → Users.user_id
+
+**Derived From:** FD20 ((build_id, user_id) → rating_build_value)
+
+**Functional Dependency Enforced:** FD20
+
+**Lossless Join Check:**
+- (build_id, user_id) is NOT a proper subset of the original composite key
+- The composite key becomes the primary key of Rating_Build
+- Intersection: I = {build_id, user_id}
+- FD20 guarantees the composite key determines the non-key attribute
+- Join with Universal_Relation_1NF on (build_id, user_id) is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD20: (build_id, user_id) → rating_build_value
+  - All attributes present ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 21: Favorite_Build
+
+Favorite_Build(build_id, user_id)
+PK: (build_id, user_id)
+FK: build_id → Build.build_id
+FK: user_id → Users.user_id
+
+**Derived From:** FD21 ((build_id, user_id) → ∅)
+
+**Functional Dependency Enforced:** FD21
+
+**Lossless Join Check:**
+- (build_id, user_id) forms the complete schema of Favorite_Build
+- Intersection: I = {build_id, user_id}
+- The composite key captures the entire relationship
+- Join with Universal_Relation_1NF on (build_id, user_id) is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD21: (build_id, user_id) → ∅
+  - The relationship-only FD is preserved through the composite key structure ✓
+  - Determinant present as primary key
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 22: Review
+
+Review(review_id, review_build_id, review_user_id, review_content, review_created_at)
+PK: review_id
+Alt Key: (review_build_id, review_user_id)
+FK: review_build_id → Build.build_id
+FK: review_user_id → Users.user_id
+
+**Derived From:** FD22 (review_id → review_build_id, review_user_id, review_content, review_created_at)
+
+**Functional Dependency Enforced:** FD22
+
+**Lossless Join Check:**
+- review_id was part of the composite key in Universal_Relation_1NF
+- Now review_id becomes the sole primary key in Review
+- Intersection: I = {review_id}
+- FD22 guarantees review_id uniquely determines all non-key attributes
+- Join with Universal_Relation_1NF on review_id is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD22: review_id → review_build_id, review_user_id, review_content, review_created_at
+  - All attributes present in Review ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+==== 2NF Relation 23: Suggestions
+
+Suggestions(suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
+PK: suggestion_id
+FK: suggestion_user_id → Users.user_id
+FK: suggestion_admin_id → Admins.admin_user_id
+
+**Derived From:** FD23 (suggestion_id → suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
+
+**Functional Dependency Enforced:** FD23
+
+**Lossless Join Check:**
+- suggestion_id was part of the composite key in Universal_Relation_1NF
+- Now suggestion_id becomes the sole primary key in Suggestions
+- Intersection: I = {suggestion_id}
+- FD23 guarantees suggestion_id uniquely determines all non-key attributes
+- Join with Universal_Relation_1NF on suggestion_id is lossless
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD23: suggestion_id → suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+  - All attributes present ✓
+  - Directly enforceable
+- **Result: PRESERVED ✓**
+
+---
+
+=== 5.3 1NF Multivalued Relations – Already in 2NF
+
+The four relations decomposed in the 1NF step (R1–R4: Case_Storage_Form_Factors, Case_Ps_Form_Factors, Case_Mobo_Form_Factors, Cooler_Cpu_Sockets) are already in 2NF because they contain only primary key attributes or attributes fully determined by their composite keys. No further decomposition is required.
+
+==== Relation R1: Case_Storage_Form_Factors
+
+Case_Storage_Form_Factors(component_id, pc_case_storage_form_factor, pc_case_storage_num_slots)
+PK: (component_id, pc_case_storage_form_factor)
+FK: component_id → Components.component_id
+
+**2NF Status:** IN 2NF ✓
+- Composite primary key (component_id, pc_case_storage_form_factor)
+- The only non-key attribute pc_case_storage_num_slots depends on the entire composite key
+- No partial dependencies exist
+
+**Lossless Join Check:**
+- Preserved from 1NF decomposition
+- Already verified as lossless in 1NF section
+- **Result: LOSSLESS ✓**
+
+**Dependency Preservation Check:**
+- FD24: (component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots
+  - All attributes present ✓
+  - Already verified as preserved in 1NF section
+- **Result: PRESERVED ✓**
+
+---
+
+==== Relation R2: Case_Ps_Form_Factors
+
+Case_Ps_Form_Factors(component_id, pc_case_ps_form_factor)
+PK: (component_id, pc_case_ps_form_factor)
+FK: component_id → Components.component_id
+
+**2NF Status:** IN 2NF ✓
+- Only key attributes; no non-key attributes to form partial dependencies
+
+**Lossless Join Check:** LOSSLESS ✓ (preserved from 1NF)
+
+**Dependency Preservation Check:** PRESERVED ✓ (FD26 preserved in 1NF)
+
+---
+
+==== Relation R3: Case_Mobo_Form_Factors
+
+Case_Mobo_Form_Factors(component_id, pc_case_mobo_form_factor)
+PK: (component_id, pc_case_mobo_form_factor)
+FK: component_id → Components.component_id
+
+**2NF Status:** IN 2NF ✓
+- Only key attributes
+
+**Lossless Join Check:** LOSSLESS ✓ (preserved from 1NF)
+
+**Dependency Preservation Check:** PRESERVED ✓ (FD25 preserved in 1NF)
+
+---
+
+==== Relation R4: Cooler_Cpu_Sockets
+
+Cooler_Cpu_Sockets(component_id, cooler_cpu_socket)
+PK: (component_id, cooler_cpu_socket)
+FK: component_id → Components.component_id
+
+**2NF Status:** IN 2NF ✓
+- Only key attributes
+
+**Lossless Join Check:** LOSSLESS ✓ (preserved from 1NF)
+
+**Dependency Preservation Check:** PRESERVED ✓ (FD27 preserved in 1NF)
+
+---
+
+== 6. Third Normal Form (3NF) Analysis
+
+=== 6.1 Transitive Dependency Analysis for Each 2NF Relation
+
+==== Relation 1: Users
+
+Users(user_id, username, password, email)
+PK: user_id
+FD: FD1 (user_id → username, password, email)
+
+**Non-key Attributes:** username, password, email
+
+**Transitive Dependency Check:**
+- Direct dependencies exist:
+  - user_id → username (direct, via FD1)
+  - user_id → password (direct, via FD1)
+  - user_id → email (direct, via FD1)
+- Potential transitive chains: None
+  - username does not determine any attribute
+  - password does not determine any attribute
+  - email does not determine any attribute
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- All non-key attributes depend directly on the primary key
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 2: Admins
+
+Admins(admin_user_id, user_id)
+PK: admin_user_id
+FD: FD2 (admin_user_id → ∅)
+
+**Non-key Attributes:** user_id
+
+**Transitive Dependency Check:**
+- user_id is a **foreign key** (reference to Users.user_id)
+- Foreign keys are not derived attributes; they are stored references to another table
+- No chain exists because:
+  - user_id is not functionally determined by any intermediate non-key attribute
+  - user_id is directly stored in the relation as a reference
+  - No further attributes depend on user_id within this relation
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- Foreign keys do not participate in transitive dependencies (they are direct references, not derived values)
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 3: Components
+
+Components(component_id, component_name, component_brand, component_price, component_type, component_img_url)
+PK: component_id
+FD: FD3 (component_id → component_name, component_brand, component_price, component_type, component_img_url)
+
+**Non-key Attributes:** component_name, component_brand, component_price, component_type, component_img_url
+
+**Transitive Dependency Check:**
+- All non-key attributes depend directly on component_id:
+  - component_id → component_name (direct)
+  - component_id → component_brand (direct)
+  - component_id → component_price (direct)
+  - component_id → component_type (direct)
+  - component_id → component_img_url (direct)
+- No attribute depends on another non-key attribute
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- All attributes are directly determined by the primary key
+- No chains of the form PK → B → A exist
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 4: Cpu
+
+Cpu(component_id, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp)
+PK: component_id
+FD: FD4 (component_id → cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp)
+
+**Non-key Attributes:** cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp
+
+**Transitive Dependency Check:**
+- All non-key attributes depend directly on component_id
+- No non-key attribute determines another non-key attribute
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 5–17: Gpu, Memory, Storage, Power_Supply, Motherboard, Pc_Case, Cooler, Memory_Card, Optical_Drive, Sound_Card, Cables, Network_Adapter, Network_Card
+
+All 13 of these relations follow the identical pattern:
+
+**Pattern:**
+```
+Relation(component_id, [component_specific_attributes])
+PK: component_id
+FD: component_id → [all dependent attributes]
+```
+
+**Transitive Dependency Check (for each):**
+- All non-key attributes depend directly on component_id
+- No attribute depends on another non-key attribute
+- No chains of the form PK → B → A exist
+
+**Conclusion for Relations 5–17:** **NO transitive dependencies exist** ✓
+- **3NF Status: ALL IN 3NF** ✓
+
+---
+
+==== Relation 18: Build
+
+Build(build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved)
+PK: build_id
+FD: FD18 (build_id → build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved)
+
+**Non-key Attributes:** build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved
+
+**Transitive Dependency Check:**
+- Direct dependencies:
+  - build_id → build_user_id (direct)
+  - build_id → build_name (direct)
+  - build_id → build_created_at (direct)
+  - build_id → build_description (direct)
+  - build_id → build_total_price (direct)
+  - build_id → build_is_approved (direct)
+- build_user_id is a **foreign key reference** (not derived)
+- No non-key attribute determines another non-key attribute
+- No chains exist:
+  - build_user_id does not determine build_name, build_created_at, etc.
+  - Each attribute stands independently, directly determined by build_id
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- All attributes depend directly on the primary key
+- Foreign key does not create transitive dependencies
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 19: Build_Component
+
+Build_Component(build_id, component_id, build_component_num_components)
+PK: (build_id, component_id)
+FD: FD19 ((build_id, component_id) → build_component_num_components)
+
+**Non-key Attributes:** build_component_num_components
+
+**Transitive Dependency Check:**
+- Only one non-key attribute exists: build_component_num_components
+- It depends directly on the composite primary key (build_id, component_id)
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- Only one non-key attribute present
+- Direct dependency on composite PK
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 20: Rating_Build
+
+Rating_Build(build_id, user_id, rating_build_value)
+PK: (build_id, user_id)
+FD: FD20 ((build_id, user_id) → rating_build_value)
+
+**Non-key Attributes:** rating_build_value
+
+**Transitive Dependency Check:**
+- Only one non-key attribute: rating_build_value
+- Depends directly on (build_id, user_id)
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- Single non-key attribute
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 21: Favorite_Build
+
+Favorite_Build(build_id, user_id)
+PK: (build_id, user_id)
+FD: FD21 ((build_id, user_id) → ∅)
+
+**Non-key Attributes:** (none)
+
+**Transitive Dependency Check:**
+- No non-key attributes exist
+- Therefore, no transitive dependencies can occur
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 22: Review
+
+Review(review_id, review_build_id, review_user_id, review_content, review_created_at)
+PK: review_id
+Alt Key: (review_build_id, review_user_id)
+FD: FD22 (review_id → review_build_id, review_user_id, review_content, review_created_at)
+
+**Non-key Attributes:** review_build_id, review_user_id, review_content, review_created_at
+
+**Transitive Dependency Check:**
+- Direct dependencies on review_id:
+  - review_id → review_build_id (direct, foreign key reference)
+  - review_id → review_user_id (direct, foreign key reference)
+  - review_id → review_content (direct)
+  - review_id → review_created_at (direct)
+- review_build_id and review_user_id are foreign keys (not derived)
+- No non-key attribute determines another non-key attribute:
+  - review_build_id does not determine review_content or review_created_at
+  - review_user_id does not determine review_content or review_created_at
+  - review_content does not determine review_created_at
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- All attributes depend directly on review_id
+- Foreign keys do not create transitive dependencies
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 23: Suggestions
+
+Suggestions(suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
+PK: suggestion_id
+FD: FD23 (suggestion_id → suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
+
+**Non-key Attributes:** suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type
+
+**Transitive Dependency Check:**
+- All non-key attributes depend directly on suggestion_id:
+  - suggestion_id → suggestion_user_id (direct, foreign key)
+  - suggestion_id → suggestion_admin_id (direct, foreign key)
+  - suggestion_id → suggestion_link (direct)
+  - suggestion_id → suggestion_admin_comment (direct)
+  - suggestion_id → suggestion_description (direct)
+  - suggestion_id → suggestion_status (direct)
+  - suggestion_id → suggestion_component_type (direct)
+- No non-key attribute determines another non-key attribute
+- No chains of form PK → B → A exist
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- All attributes depend directly on the primary key
+- Foreign keys are not derived values
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relation 24: Case_Storage_Form_Factors
+
+Case_Storage_Form_Factors(component_id, pc_case_storage_form_factor, pc_case_storage_num_slots)
+PK: (component_id, pc_case_storage_form_factor)
+FD: FD24 ((component_id, pc_case_storage_form_factor) → pc_case_storage_num_slots)
+
+**Non-key Attributes:** pc_case_storage_num_slots
+
+**Transitive Dependency Check:**
+- Only one non-key attribute: pc_case_storage_num_slots
+- Depends directly on composite PK (component_id, pc_case_storage_form_factor)
+
+**Conclusion:** **NO transitive dependencies exist** ✓
+- **3NF Status: IN 3NF** ✓
+
+---
+
+==== Relations 25–27: Case_Ps_Form_Factors, Case_Mobo_Form_Factors, Cooler_Cpu_Sockets
+
+All three relations follow the identical pattern:
+
+**Pattern:**
+```
+Relation(component_id, [attribute])
+PK: (component_id, [attribute])
+FD: ([attributes] → ∅)
+```
+
+**Transitive Dependency Check (for each):**
+- No non-key attributes exist
+- Therefore, no transitive dependencies can occur
+
+**Conclusion for Relations 25–27:** **NO transitive dependencies exist** ✓
+- **3NF Status: ALL IN 3NF**
+
+|| Relation =||= PK Type =||= Non-Key Attributes =||= Transitive Dependency? =||= 3NF Status =||
+|| Users || Single (user_id) || 3 attributes || NO || ✓ IN 3NF ||
+|| Admins || Single (admin_user_id) || 1 FK || NO || ✓ IN 3NF ||
+|| Components || Single (component_id) || 5 attributes || NO || ✓ IN 3NF ||
+|| Cpu–Network_Card (14 relations) || Single (component_id) || Variable || NO || ✓ IN 3NF ||
+|| Build || Single (build_id) || 6 attributes || NO || ✓ IN 3NF ||
+|| Build_Component || Composite || 1 attribute || NO || ✓ IN 3NF ||
+|| Rating_Build || Composite || 1 attribute || NO || ✓ IN 3NF ||
+|| Favorite_Build || Composite || 0 attributes || NO || ✓ IN 3NF ||
+|| Review || Single (review_id) || 4 attributes || NO || ✓ IN 3NF ||
+|| Suggestions || Single (suggestion_id) || 7 attributes || NO || ✓ IN 3NF ||
+|| Case_Storage_Form_Factors || Composite || 1 attribute || NO || ✓ IN 3NF ||
+|| Case_Ps_Form_Factors || Composite || 0 attributes || NO || ✓ IN 3NF ||
+|| Case_Mobo_Form_Factors || Composite || 0 attributes || NO || ✓ IN 3NF ||
+|| Cooler_Cpu_Sockets || Composite || 0 attributes || NO || ✓ IN 3NF ||
+
+== 6. Final BCNF Decomposed Schema
+
+=== 6.1 Complete List of All Relations
+
+1. **Users**(user_id, username, password, email)
+   - PK: user_id
+
+2. **Admins**(admin_user_id)
+   - PK: admin_user_id
+   - FK: admin_user_id → Users.user_id
+
+3. **Components**(component_id, component_name, component_brand, component_price, component_type, component_img_url)
+   - PK: component_id
+
+4. **Cpu**(component_id, cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+5. **Gpu**(component_id, gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+6. **Memory**(component_id, memory_type, memory_speed, memory_capacity, memory_modules)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+7. **Storage**(component_id, storage_type, storage_capacity, storage_form_factor)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+8. **Power_Supply**(component_id, power_supply_type, power_supply_wattage, power_supply_form_factor)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+9. **Motherboard**(component_id, motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots)
+   - PK: component_id
+   - FK: component_id → Components.component_id
+
+10. **Pc_Case**(component_id, pc_case_cooler_max_height, pc_case_gpu_max_length)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+11. **Cooler**(component_id, cooler_type, cooler_height, cooler_max_tdp_supported)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+12. **Memory_Card**(component_id, memory_card_num_slots, memory_card_interface)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+13. **Optical_Drive**(component_id, optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+14. **Sound_Card**(component_id, sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+15. **Cables**(component_id, cables_length_cm, cables_type)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+16. **Network_Adapter**(component_id, network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+17. **Network_Card**(component_id, network_card_num_ports, network_card_speed, network_card_interface)
+    - PK: component_id
+    - FK: component_id → Components.component_id
+
+18. **Build**(build_id, build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved)
+    - PK: build_id
+    - FK: build_user_id → Users.user_id
+
+19. **Build_Component**(build_id, component_id, build_component_num_components)
+    - PK: (build_id, component_id)
+    - FK: build_id → Build.build_id
+    - FK: component_id → Components.component_id
+
+20. **Rating_Build**(build_id, user_id, rating_build_value)
+    - PK: (build_id, user_id)
+    - FK: build_id → Build.build_id
+    - FK: user_id → Users.user_id
+
+21. **Favorite_Build**(build_id, user_id)
+    - PK: (build_id, user_id)
+    - FK: build_id → Build.build_id
+    - FK: user_id → Users.user_id
+
+22. **Review**(review_id, review_build_id, review_user_id, review_content, review_created_at)
+    - PK: review_id
+    - Alt Key: (review_build_id, review_user_id)
+    - FK: review_build_id → Build.build_id
+    - FK: review_user_id → Users.user_id
+
+23. **Suggestions**(suggestion_id, suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type)
+    - PK: suggestion_id
+    - FK: suggestion_user_id → Users.user_id
+    - FK: suggestion_admin_id → Admins.admin_user_id
+
+24. **Case_Storage_Form_Factors**(component_id, pc_case_storage_form_factor, pc_case_storage_num_slots)
+    - PK: (component_id, pc_case_storage_form_factor)
+    - FK: component_id → Components.component_id
+
+25. **Case_Ps_Form_Factors**(component_id, pc_case_ps_form_factor)
+    - PK: (component_id, pc_case_ps_form_factor)
+    - FK: component_id → Components.component_id
+
+26. **Case_Mobo_Form_Factors**(component_id, pc_case_mobo_form_factor)
+    - PK: (component_id, pc_case_mobo_form_factor)
+    - FK: component_id → Components.component_id
+
+27. **Cooler_Cpu_Sockets**(component_id, cooler_cpu_socket)
+    - PK: (component_id, cooler_cpu_socket)
+    - FK: component_id → Components.component_id
+
+