Question

21. Let A and B be two sets. Prove that: (A - B) U B = A if and only if B is a subset of A.
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Answer 1

Answer:

(A-B) U B = A U B you need to show two things:

a) (A-B)UB is a subset of AUB and

b) AUB is a subset of (A-B)UB.

To show a), let x ε (A-B)UB.

Then x ε A-B or x ε B

If x ε A-B then x ε A and x ε B', from which is follows that x ε AUB

If x ε B then x ε AUB, from which it follows that x ε AUB

Therefore (A-B)UB is a subset of AUB

To show b), let x ε AUB

Then, x ε A or x ε B.

If x ε A then x ε A-B, from which it follows that x ε (A-B)UB

If x ε B then x ε AUB

Therefore, AUB is a subset of (A-B)UB

This proves that (A-B)UB = AUB.

Answer 2

Answer:

Step-by-step explanation:

We have to prove (A-B)UB=A

SO we can write (A-B) as A intersection B'

So it will be, (A intersection B')U B

So (AUB) intersection(B'UB)

So A intersection (AUB)

So A intersection A =A