1. ## proof

If A is a subset of B, then A X A is a subset of B X B

hoe do you prove this

If A is a subset of B, then A X A is a subset of B X B

hoe do you prove this
$\displaystyle A\subseteq B$ so if $\displaystyle x\in A$ then $\displaystyle x\in B$

A X A=$\displaystyle \{ (x,y) | x,y\in A \}$ but since $\displaystyle x,y\in A$, $\displaystyle x$ and $\displaystyle y \in B$

So all $\displaystyle (x,y)\in B$, so AXA$\displaystyle \subseteq$BXB

3. how do you prove the reverse of that

if A X A subset B X B, then A subset of B

how do you prove the reverse of that
if A X A subset B X B, then A subset of B
If $\displaystyle x\in A$ then $\displaystyle (x,x)\in A\times A$. Can you explain that?

Then can you prove that implies that $\displaystyle x\in B?$

How does that prove that $\displaystyle A \subseteq B?$

5. Proof: Assume x is an element of A. (x,x) is therefore an element of A X A because the definition of A X A is the set of (x,x) such that x is an element of A and x is an element of A. Since , A X A is a subset of B X B (x,x) must also be an element of B X B which means X is an element of B and x is an element of B. x is an element of B so A is a subset of B

is this right reasoning