= Normalization

== 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,
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_form_factor, pc_case_storage_num_slots,
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_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
)

== 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,
  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–FD23 (all except FD24–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 ✓
- 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)
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) || 0 attributes || 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 ||

== BCNF Analysis

|| Relation =||= Primary Key =||= FD =||= Determinant =||= Is Determinant a CK? =||= Dependent Attributes =||= BCNF Satisfied =||
|| Cpu || component_id || FD4 || component_id || YES (PK) || cpu_socket, cpu_cores, cpu_threads, cpu_base_clock, cpu_boost_clock, cpu_tdp || ✓ YES ||
|| Gpu || component_id || FD5 || component_id || YES (PK) || gpu_vram, gpu_tdp, gpu_base_clock, gpu_boost_clock, gpu_chipset, gpu_length || ✓ YES ||
|| Memory || component_id || FD6 || component_id || YES (PK) || memory_type, memory_speed, memory_capacity, memory_modules || ✓ YES ||
|| Storage || component_id || FD7 || component_id || YES (PK) || storage_type, storage_capacity, storage_form_factor || ✓ YES ||
|| Power_Supply || component_id || FD8 || component_id || YES (PK) || power_supply_type, power_supply_wattage, power_supply_form_factor || ✓ YES ||
|| Motherboard || component_id || FD9 || component_id || YES (PK) || motherboard_socket, motherboard_chipset, motherboard_form_factor, motherboard_ram_type, motherboard_num_ram_slots, motherboard_max_ram_capacity, motherboard_pci_express_slots || ✓ YES ||
|| Pc_Case || component_id || FD10 || component_id || YES (PK) || pc_case_cooler_max_height, pc_case_gpu_max_length || ✓ YES ||
|| Cooler || component_id || FD11 || component_id || YES (PK) || cooler_type, cooler_height, cooler_max_tdp_supported || ✓ YES ||
|| Memory_Card || component_id || FD12 || component_id || YES (PK) || memory_card_num_slots, memory_card_interface || ✓ YES ||
|| Optical_Drive || component_id || FD13 || component_id || YES (PK) || optical_drive_form_factor, optical_drive_type, optical_drive_interface, optical_drive_write_speed, optical_drive_read_speed || ✓ YES ||
|| Sound_Card || component_id || FD14 || component_id || YES (PK) || sound_card_sample_rate, sound_card_bit_depth, sound_card_chipset, sound_card_interface, sound_card_channel || ✓ YES ||
|| Cables || component_id || FD15 || component_id || YES (PK) || cables_length_cm, cables_type || ✓ YES ||
|| Network_Adapter || component_id || FD16 || component_id || YES (PK) || network_adapter_wifi_version, network_adapter_interface, network_adapter_num_antennas || ✓ YES ||
|| Network_Card || component_id || FD17 || component_id || YES (PK) || network_card_num_ports, network_card_speed, network_card_interface || ✓ YES ||

|| Users || user_id || FD1 || user_id || YES (PK) || username, password, email || ✓ YES ||
|| Admins || admin_user_id || FD2 || admin_user_id || YES (PK) || (none) || ✓ YES ||
|| Components || component_id || FD3 || component_id || YES (PK) || component_name, component_brand, component_price, component_type, component_img_url || ✓ YES ||
|| Build || build_id || FD18 || build_id || YES (PK) || build_user_id, build_name, build_created_at, build_description, build_total_price, build_is_approved || ✓ YES ||
|| Review || review_id || FD22 || review_id || YES (PK) || review_build_id, review_user_id, review_content, review_created_at || ✓ YES ||

|| Build_Component || (build_id, component_id) || FD19 || (build_id, component_id) || YES (PK) || build_component_num_components || ✓ YES ||
|| Rating_Build || (build_id, user_id) || FD20 || (build_id, user_id) || YES (PK) || rating_build_value || ✓ YES ||
|| Favorite_Build || (build_id, user_id) || FD21 || (build_id, user_id) || YES (PK) || (none) || ✓ YES ||
|| Suggestions || suggestion_id || FD23 || suggestion_id || YES (PK) || suggestion_user_id, suggestion_admin_id, suggestion_link, suggestion_admin_comment, suggestion_description, suggestion_status, suggestion_component_type || ✓ YES ||

|| Case_Storage_Form_Factors || (component_id, pc_case_storage_form_factor) || FD24 || (component_id, pc_case_storage_form_factor) || YES (PK) || pc_case_storage_num_slots || ✓ YES ||
|| Case_Ps_Form_Factors || (component_id, pc_case_ps_form_factor) || FD26 || (component_id, pc_case_ps_form_factor) || YES (PK) || (none) || ✓ YES ||
|| Case_Mobo_Form_Factors || (component_id, pc_case_mobo_form_factor) || FD25 || (component_id, pc_case_mobo_form_factor) || YES (PK) || (none) || ✓ YES ||
|| Cooler_Cpu_Sockets || (component_id, cooler_cpu_socket) || FD27 || (component_id, cooler_cpu_socket) || YES (PK) || (none) || ✓ YES ||

== 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