Changes between Version 34 and Version 35 of Normalization

Timestamp:

05/04/26 11:39:14 (3 weeks ago)

Author:

221296

Comment:

--

Normalization

@@ -294 +294 @@
 {{{Lossless join:}}}
 [[BR]]
-The decomposition is lossless because each new relation is formed around a determinant that becomes the key of that relation. The common attributes used in the decomposition are keys in at least one of the resulting relations, so no information is lost.
+To verify that the decomposition is lossless, we use the formal condition:
+
+For every decomposition of R into relations Ri and Rj, the intersection of their attributes must functionally determine at least one of the relations:
+
+Ri ∩ Rj → Ri  OR  Ri ∩ Rj → Rj
+
+In this decomposition, each relation is formed based on a determinant (such as user_id, plan_id, course_id, etc.), which becomes the primary key of that relation.
+
+The shared attributes between relations (e.g. user_id, course_id, subscription_id) are keys in at least one of the involved relations.
+
+Therefore, the condition for lossless join is satisfied, and the decomposition is lossless with respect to the original universal relation R.
 
 
 {{{Dependency preservation:}}}
 [[BR]]
-The decomposition is dependency-preserving because each functional dependency FD1–FD15 is represented fully inside one of the resulting relations. CourseCategory has only its composite key and no additional non-trivial functional dependencies.
+To verify dependency preservation, we check whether the union of the projections of the functional dependencies onto each resulting relation is equivalent to the original set of functional dependencies:
+
+(⋃ projection of F onto Ri)+ = F+
+
+In this decomposition, each functional dependency from the initial set (FD1–FD15) is fully contained within at least one of the resulting relations.
+
+For example:
+- user_id → user attributes is preserved in UserEntity
+- course_id → course attributes is preserved in Course
+- subscription_id → subscription attributes is preserved in UserSubscription
+
+Since no functional dependency is lost and all can be enforced locally within the resulting relations, the decomposition is dependency-preserving.
 
 == Check for 3NF (and decomposition)