= Proof of Candidate Key =

== Definition ==

A set of attributes X is a candidate key if:

X⁺ = R
X is minimal

The closure X⁺ is computed by repeatedly applying functional dependencies until no additional attributes can be derived.

== Step 1 – Testing Single Attribute Candidate Keys ==

=== Attempt 1 ===

K = {attendance_id}

=== Closure ===

(attendance_id)⁺ gives:

attendance_status
table_number
guest_id
event_id

Using additional dependencies:

event_id → event_type, wedding_id
guest_id → guest_first_name, guest_last_name, rsvp_status, wedding_id
wedding_id → date, budget, notes, user_id
user_id → first_name, last_name, email, phone_number, gender, birthday

=== Missing Dependencies ===

The closure still does NOT determine:

venue booking information
photographer booking information
band booking information
registrar booking information
priest and church information

Therefore:

(attendance_id)⁺ ≠ R

Conclusion:

attendance_id is NOT a candidate key.

== Step 2 – Testing Two Attributes ==

K = {attendance_id, priest_id}

=== Closure ===

Additional attributes derived:

priest_name
priest_contact
church_id
church_name
church_location

=== Missing Dependencies ===

Still missing:

venue booking information
photographer booking information
band booking information
registrar booking information

Therefore:

(attendance_id, priest_id)⁺ ≠ R

Conclusion:

K is NOT a candidate key.

== Step 3 – Testing Three Attributes ==

K = {
attendance_id,
venue_booking_id,
photographer_booking_id
}

=== Closure ===

Additional attributes derived:

venue information
photographer information
wedding information
user information

=== Missing Dependencies ===

Still missing:

band booking information
registrar booking information
priest/church information

Therefore:

K⁺ ≠ R

Conclusion:

K is NOT a candidate key.

== Step 4 – Testing Larger Attribute Sets ==

K = {
attendance_id,
priest_id,
venue_booking_id,
band_booking_id
}

=== Missing Dependencies ===

Still missing:

photographer booking information
registrar booking information

Therefore:

K⁺ ≠ R

Conclusion:

K is NOT a candidate key.

== Step 5 – Including All Booking Branches ==

K = {
attendance_id,
venue_booking_id,
photographer_booking_id,
band_booking_id,
registrar_booking_id
}

=== Missing Dependencies ===

Still missing:

priest information
church information

Therefore:

K⁺ ≠ R

Conclusion:

K is NOT a candidate key.

== Step 6 – Final Candidate Key ==

K = {
attendance_id,
venue_booking_id,
photographer_booking_id,
band_booking_id,
registrar_booking_id,
priest_id
}

=== Closure ===

Using all functional dependencies:

attendance branch is determined
event branch is determined
guest branch is determined
wedding branch is determined
venue booking branch is determined
photographer booking branch is determined
band booking branch is determined
registrar booking branch is determined
priest branch is determined
church branch is determined

No attributes remain undetermined.

Therefore:

K⁺ = R

Conclusion:

K is a candidate key.

== Minimality Proof ==

A candidate key must be minimal.

Removing any attribute from K causes loss of at least one independent entity branch:

|| Removed Attribute || Lost Information ||
|| attendance_id || attendance branch ||
|| venue_booking_id || venue booking branch ||
|| photographer_booking_id || photographer booking branch ||
|| band_booking_id || band booking branch ||
|| registrar_booking_id || registrar booking branch ||
|| priest_id || priest/church branch ||

Therefore every attribute in K is necessary.

== Why church_id Is Not Included ==

Suppose we define:

K = {
attendance_id,
venue_booking_id,
photographer_booking_id,
band_booking_id,
registrar_booking_id,
priest_id,
church_id
}

This violates minimality because:

priest_id → church_id

church_id is already derivable from priest_id.

Therefore church_id is redundant.

The resulting set would be a superkey, NOT a candidate key.

== Final Candidate Key ==

K = {
attendance_id,
venue_booking_id,
photographer_booking_id,
band_booking_id,
registrar_booking_id,
priest_id
}

This key:

Determines all attributes of R
Is minimal
Satisfies the formal definition of a candidate key