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 $\displaystyle x \in p$.

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

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

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

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

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.