Question #12398

prove (A-B) Union B = A iff B contained in A

Expert's answer

Let'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 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. As B belongs to A then AUB = A and so, (A-B)UB = A.

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

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