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dlinn [17]
3 years ago
15

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

iversal 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.
Mathematics
1 answer:
tekilochka [14]3 years ago
5 0

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.

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