Set Theory Identities: A = B if A, B, and C satisfy key set relations

[Stevo6754]

Homework Statement

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

A \cup C = B \cup C and A \cap C = B \cap C

Homework Equations

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

A) A \cup C = B \cup 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 \cup C = C = B \cup C
but A \neq B

B) A \cap C = B \cap C

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

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 \cup C = B \cup C, this implies that (A \cup C) \subseteq (B \cup C), and (B \cup C) \subseteq
(A \cup C)

So, (A \cup C) \subseteq (B \cup C)

Same goes for (A \cap C) \subseteq ( B \cap C),

In order to prove A = B I need to prove A \subseteq B and B \subseteq A.

So I have these premises and a conclusion, but I am 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!

 

[Hurkyl]

I don't think it's too hard to prove this by looking at elements in combination with algebraic identities, rather than a purely algebraic proof, if you are inclined to do so.


Rather than trying to prove that A=B, you may find it easier constructing a set that measures how different A and B are, and then proving something about that.


What identities are you considering using? Rewriting equality in terms of subsets is a place to start, but you don't seem to have invoked any properties of union and intersection yet.

I imagine you're probably using a list like the one here, along with the ones in the next section relating meet and join to ordering.

(In your case, the set-theoretic symbols \cap, \cup, \subseteq correspond to the lattice algebra symbols \wedge, \vee, \leq)

FYI, that list of identities is not enough. You need these as well.

You may find identities relating to complementation useful too.