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\\otin C`

Therefore,
`x\in( A\cap B)` but
`x\\otin (A\cap C)`

Hence, it is true.

Conversely , Let
`x\in(A\cap B)` but
`x\\otin(A\cap C)`

Then
`x\in A` and
`x\in B`

When
`x\\otin ( A\cap C)` then
`x\\otin C`

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

Hence, the statement is true.