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.