Give a proof or a counterexample for each of the following?

Question

Give a proof or a counterexample for each of the following statements: a. A – (B ∩ A) = A – B. b. A ⊕A = A.

Answer 1

The first is true: A – (B ∩ A) = A ∩ (B ∩ A)', where ' means the complement. By De Morgan, A – (B ∩ A) = A ∩ (B' U A') = (A ∩ B') U (A ∩ A)' = (A - B) U { } = A - B).

Observe that A - B = {x | x is in A and x is not in B} = A ∩ B' . This definition makes sense even if A and B are disjoint. In this case, A - B = A and A – (B ∩ A = A. This is no counter example.

The second is false. A ⊕A = {a1 + a2 | a1 and a2 are in A}. If A = [1,2], then A ⊕A = [2,4], different from A.

Answer 2

Both are true.
a. Steiner's answer is correct.
b. A U A = A. proof: x E A U A ↔ x E A or x E A ↔ x E A.

Answer 3

Counter example for the first one:
A = set of even integers
B = set of odd integers

B ∩ A = empty set
A - (B ∩ A) = A
A - B is undefined or makes no sense


A – (B ∩ A) = A – B iff (if and only if)
B ∩ A = B
ie B is a subset of A

Answer 4

those are notably basic. once you're ever going to income the textile, engaged on questions like those your self is a thank you to do it. right this is a tremendous hint; the rest truly could desire to be as much as you. (a) via (d) are fake. (e) is actual.