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Answer to Question #12398 in Real Analysis for shivakumar

Question #12398
prove (A-B) Union B = A iff B contained in A
Let&#039;s prove that (A-B) U B = A U B. We 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 &epsilon; (A-B)UB.

Then x &epsilon; A-B or x &epsilon; B

If x &epsilon; A-B then x &epsilon; A and x &epsilon; B&#039;, from which is follows that x &epsilon; AUB
If x &epsilon; B then x &epsilon; AUB, from which it follows that x &epsilon; AUB

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

To show b), let x &epsilon; AUB

Then, x &epsilon; A or x &epsilon; B.

If x &epsilon; A then x &epsilon; A-B, from which it follows that x &epsilon; (A-B)UB
If x &epsilon; B then x &epsilon; AUB

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

This proves that (A-B)UB = AUB. As B belongs to A then AUB = A and so, (A-B)UB = A.

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