Question

Prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For or all sets A, B, and C, A∩(B−C) = (A∩ B)−(A∩C)

Hint: The statement is true. Sketch of proof: If x ∈ A∩(B−C), then x∈A and x∈B and x∉C. So it is true that x∈A and x∈B and that x∈A and x∉C.Conversely, if x ∈ (A∩ B)−(A∩C), then x∈A and x∈B, but x ∉ A∩C, and so x∉C.

Answer 1

Answer with Step-by-step explanation:

We are given that A, B and C are subsets of universal set U.

We have to prove that

$A\cap (B-C)=(A\cap B)-(A\cap C)$

Proof:

Let x$\in A\cap (B-C)$

Then $x\in A$ and $x\in(B-C)$

When $x\in ( B-C)$then $x\in B$ but $x\notin C$

Therefore, $x\in( A\cap B)$ but $x\notin (A\cap C)$

Hence, it is true.

Conversely , Let $x\in(A\cap B)$ but $x\notin(A\cap C)$

Then $x\in A$ and $x\in B$

When $x\notin ( A\cap C)$ then $x\notin C$

Therefor,$x\in A\cap (B-C)$

Hence, the statement is true.