Changes between Version 2 and Version 3 of Proof of Candidate Key

Timestamp:

05/18/26 21:10:19 (8 days ago)

Author:

211171

Comment:

--

Proof of Candidate Key

@@ -215 +215 @@
 Therefore every attribute in K is necessary.
 
+== Verification of Alternative Candidate Keys ==
+
+A relation may contain multiple candidate keys.
+Therefore, after identifying one candidate key, we must verify whether another minimal attribute set can also determine all attributes of R.
+
+=== Testing Possible Alternative Key ===
+
+The only realistic possibility is replacing:
+
+priest_id
+
+with:
+
+church_id
+
+because of the dependency:
+
+priest_id → church_id
+
+However, the reverse dependency does NOT hold:
+
+church_id ↛ priest_id
+
+A church may have multiple priests, therefore church_id alone cannot uniquely determine priest information.
+
+=== Alternative Key Attempt ===
+
+K₂ = {
+attendance_id,
+venue_booking_id,
+photographer_booking_id,
+band_booking_id,
+registrar_booking_id,
+church_id
+}
+
+=== Closure ===
+
+Using the functional dependencies:
+
+attendance branch is determined
+venue booking branch is determined
+photographer booking branch is determined
+band booking branch is determined
+registrar booking branch is determined
+church branch is determined
+
+However, the closure cannot derive:
+
+priest_id
+priest_name
+priest_contact
+
+because:
+
+church_id ↛ priest_id
+
+Therefore:
+
+K₂⁺ ≠ R
+
+Conclusion:
+
+K₂ is NOT a candidate key.
+
+=== Verification of Other Possible Replacements ===
+
+No other attribute can replace any component of the identified key because each attribute determines an independent branch of the relation.
+
+|| Attribute || Determines ||
+|| attendance_id || attendance/event/guest 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 ||
+
+Removing or replacing any attribute causes loss of information that cannot be derived from the remaining attributes.
+
 == Why church_id Is Not Included ==
 
@@ -255 +334 @@
 Is minimal
 Satisfies the formal definition of a candidate key
+
+No second candidate key exists because no alternative minimal attribute combination functionally determines all attributes of R.