# Operations/Proofs with Sets

Please show these proofs in great detail with all steps explained as they will serve as a template for future proofs.

1. Suppose A, B, and C are sets with A?B?C = 0. Prove or disprove: |AUBUC|= |A|+|B|+|C|.

2. Suppose A, B, and C are sets. Prove or disprove: AUB= A?B if and only if A=B.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

1. Suppose A, B and C are sets with A?B?C=0 prove or disprove that |AUBUC|= |A|+|B|+|C|

The statement is false. Here I provide with you a counter example.

A={1, 2, 3, 4}, B={0, 1, 2}, C={0, 3, 4}

Clearly, there is no element in common for the three sets A, B and C. So, A?B?C=Ã˜.

Now we can find the union of three sets A, B and C, which contains the elements 0, 1, 2, 3, and 4. Namely, AUBUC={0,1,2,3,4}.

As there are 5 elements in A?B?C, the size of AUBUC is 5. So, |AUBUC|= 5.

However, as A contains 4 elements, B has three elements, and C has three elements, |A|=4, |B|=3, and |C|=3.

So, |A|+|B|+|C|=4+3+3=10.

It is clear that |AUBUC| ? |A|+|B|+|C|

2. Let A and B be sets. Prove or disprove that AUB = A?B if and only if A=B.

Conclusion: The statement "AUB = A?B if and only if A=B" is true.

Proof: Since the above statement is an "if and only if" statement, we need to prove both sufficiency and necessity.

Sufficiency: We need to show "If A=B, then AUB = A?B"

[Proof] If A=B, then

AUB = A and A?B = A. So, AUB = A?B.

Necessity: We need to prove "If AUB = A?B, then A=B."

First of all, A ? AUB because AUB is the union of A and B.

Secondly, A?B?A because A?B is the intersection of A and B.

So, A?B?A?A?B...........................(1)

As AUB = A?B, (1) implies that A?B = A = AUB.

Similarly,

B?AUB because AUB is the union of A and B;

A?B?B because A?B is the intersection of A and B.

So, A?B?B?AUB.................................(2)

As AUB = A?B, (2) implies that A?B = B = AUB.

So, we have proved that if AUB = A?B, then A = B = A?B = AUB

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