Question

Prove the absorption law A U (ANB) = A by universal generalization (applied twice).

Answer 1

The absorption law A U (A ∩ B) = A is proven using universal generalization by showing that A U (A ∩ B) is both a subset of A and that A is a subset of A U (A ∩ B), thereby demonstrating their equivalence.

Step-by-step explanation:

To prove the absorption law A U (A ∩ B) = A using universal generalization, we'll show that the set A unioned with the intersection of A and B is equivalent to the set A alone. This is a principle in set theory, part of mathematical logic.

1. Consider any element x. If x is in A, then it's obviously in A U (A ∩ B), because A is a subset of the union.
2. If x is in (A ∩ B), then it must also be in A and B simultaneously. However, since it's in A, it's again in the union.
3. Thus, every element in A U (A ∩ B) must also be in A, showing that A U (A ∩ B) is a subset of A.
4. It is also trivial that A is a subset of A U (A ∩ B) because every set is a subset of itself unioned with any other set.
5. Since A is a subset of A U (A ∩ B) and A U (A ∩ B) is a subset of A, we can conclude that A is equal to A U (A ∩ B).