(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Can you conclude that A = B if A, B, and C are sets such that

A [tex]\cup[/tex] C = B [tex]\cup[/tex] C and A [tex]\cap[/tex] C = B [tex]\cap[/tex] C

2. Relevant equations

The above is part c of a problem. The problems a and b are as follows

A) A [tex]\cup[/tex] C = B [tex]\cup[/tex] C

My answer: I gave a counter example such that A = {a, b, c}, B = {c, d, e}

and C = {a, b, c, d, e}, thus A [tex]\cup[/tex] C = C = B [tex]\cup[/tex] C

but A [tex]\neq[/tex] B

B) A [tex]\cap[/tex] C = B [tex]\cap[/tex] C

My answer: I gave the counter example where A = {a, b, c}, B = {c, d, e}, C = {c}

So, A [tex]\cap[/tex] C = C = B [tex]\cap[/tex] C but A [tex]\neq[/tex] B

3. The attempt at a solution

Ok for this part c I could not think of a counter example. I believe they want me to use set identities. I'm honestly not sure where to begin but Ill tell you what I have in mind so far.

If A [tex]\cup[/tex] C = B [tex]\cup[/tex] C, this implies that (A [tex]\cup[/tex] C) [tex]\subseteq[/tex] (B [tex]\cup[/tex] C), and (B [tex]\cup[/tex] C) [tex]\subseteq[/tex]

(A [tex]\cup[/tex] C)

So, (A [tex]\cup[/tex] C) [tex]\subseteq[/tex] (B [tex]\cup[/tex] C)

Same goes for (A [tex]\cap[/tex] C) [tex]\subseteq[/tex] ( B [tex]\cap[/tex] C),

In order to prove A = B I need to prove A [tex]\subseteq[/tex] B and B [tex]\subseteq[/tex] A.

So I have these premises and a conclusion, but Im honestly not sure how to set this up. I'm pretty sure I need to use set identities.. If anyone has any advice to get me moving here I'd greatly appreciate it, thanks!

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# Homework Help: Set Theory Identities

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