Question

Let A and B be sets. A.) Prove the complement law A∩A ^c=∅ using a set-membership table. B.) Prove (A∩B)∪B^c =A∪B ^C

using the set theory laws.

Answer 1

The complement law A∩Ac=∅ is proven by understanding that no element can be in a set and its complement at the same time. The law (A∩B)∪Bc=A∪Bc is proven using the distributive law of set theory.

Step-by-step explanation:

Proof of Set Complement Law

To prove the complement law A∩Ac=∅, let's consider the definition of intersection and complement. The intersection of any set A with its complement Ac is the set containing all elements that are both in A and not in A at the same time, which is impossible. Therefore, by definition, the intersection is the empty set ∅.

Proof of Set Union/Intersection Law

To prove the law (A∩B)∪Bc=A∪Bc, we rely on the principles of set theory. Using the distributive law:

• A∩(B∪Bc) = A since B∪Bc is the universal set.
• A∪Bc = (A∩B)∪(A∩Bc)∪Bc by the distributive law.
• (A∩B)∪(A∩Bc)∪Bc simplifies to (A∩B)∪Bc since A∩Bc is a subset of Bc.
• Thus, (A∩B)∪Bc = A∪Bc.