Proper Subset and Cartesian Product

Question

I am trying to prove two separate claims (assuming they are true; I believe they are both true but I'm not 100% certain) but I think the proofs would be quite similar so I'm going to put them in the same post. The first is:

Given sets X, Y and Z, determine whether the following claim is true (if so, prove it):

If X ⊂ Y, then X × Z ⊂ Y × Z.

I am trying to provide a formal proof. My reasoning so far is something along the lines of: Since X ⊂ Y, every member of X is a member of Y and there is a member of Y that is not a member of X, call it a. If every member of X is a member of Y, then every member of X × Z will be a member of Y × Z. In addition, there will be another element of Y × Z, that is not in X × Z (since a is not in X). Therefore, X × Z ⊂ Y × Z.

The second claim is:

Given sets W, X, Y and Z, determine whether the following claim is true (if so, prove it):

If W × Z ⊂ X × Y, then W ⊂ X or Z ⊂ Y.

Since W × Z ⊂ X × Y, every member of W × Z is a member of X × Y and there is a member of X × Y, call it , that is not a member of W × Z. Since is not a member of W × Z, there are two possibilities: (i) x is not a member of W, which would make W ⊂ X (since every member of W is in X but x is not in W) or (ii) y is not a member of Z, which would make Z ⊂ Y (for the same reason). In either case, W ⊂ X or Z ⊂ Y.

I just want to double check that my logic is alright here and that there are no mistakes in my proofs (I am fairly new to doing proofs and would like to make sure I am learning how to do them properly). So any comments, tips or verifications would be much appreciated.

Thanks a lot!

Answer 1

Nice work!

The first claim is correct and its proof is correct as well.

The second claim is correct as well, and its proof is correct, but the statment could get more precise:

If $W\times Z\subset X\times Y$, then $W\subseteq Z$ and $X\subseteq Y$ and ($W\neq Z$ or $X\neq Z$).

Or, in the spirit of what you wrote:

If $W\times Z\subset X\times Y$, then $W\subseteq Z$ and $X\subseteq Y$ and ($W\subset Z$ or $X\subset Z$).

Looking at your proof, it seems that you had that in mind though.