Discrete Math / Set Theory - Proof

Question

Hi I have 2 problems that I need to say whether or not they are true or not and then write a proof for my answer. I am really struggling and could use a little help thank you

 

1)

 

     (X ⊆ Y / X×X ⊆ X×Y)

 

 

2)

 

     (X ⊆ Y / X×X ⊆ Y×Y)

 

 

 

 

 

Answer 1

I'll walk through #1 and see if that gives you enough to answer #2 yourself.  I'm assuming the symbol ( ----- ) is supposed to be ↔ (if and only if).  So let's address the forward direction first.  

 

Assume that X⊆Y.  Then we need to show that for any (a,b) ∈ X×X, (a,b) ∈ X×Y.  Let (a,b) ∈ X×X.  Then a,b ∈ X, which by our assumption means that b ∈ Y, which then means that (a,b) ∈ X×Y.  Therefore, X×X ⊆ X×Y

 

Conversely, assume that X×X ⊆ X×Y.  Then we need to show that for any a ∈ X, a ∈ Y.  Let a ∈ X.  Then (a,a) ∈ X×X, which implies that (a,a) ∈ X×Y.  But this can only be true if a ∈ Y.  Therefore, X ⊆ Y.

 

The approach to #2 will be similar.  Let me know if you need additional help with that.