1. ## Proofs

Let p,r,q be sets. Prove that if p is a subset of r and r is a proper subset of q, then p is a proper subset of q

2. Originally Posted by sderosa518
Let p,r,q be sets. Prove that if p is a subset of r and r is a proper subset of q, then p is a proper subset of q
Consider an element $x \in p$.

As $p \subset r$ it follows by definition of a subset that $x \in r$.

But as $r \subset q$, it follows that if $x \in r$ then $x \in q$.

So, given that $x \in p$ we have shown that $x \in q$ as well.

That is all you need to do to show that $p \subset q$.

3. But you still need to prove that p is a proper subset of q! That is, that there exist a least one member of q that is not a member of p. To do that use the fact that r is a proper subset of q. Notice that the problem does NOT say that p is a proper subset of r!

4. Originally Posted by HallsofIvy
But you still need to prove that p is a proper subset of q! That is, that there exist a least one member of q that is not a member of p. To do that use the fact that r is a proper subset of q. Notice that the problem does NOT say that p is a proper subset of r!
Good call. Care to finish the proof off then? I confess my brain's had a battering tonight.