Explore BrainMass

Operations/Proofs with Sets

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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.

https://brainmass.com/math/combinatorics/operations-proofs-with-sets-506541

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

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!