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

hoe do you prove this

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- Oct 21st 2009, 01:01 PMleinadwerdnaproof
If A is a subset of B, then A X A is a subset of B X B

hoe do you prove this - Oct 21st 2009, 01:14 PMartvandalay11
$\displaystyle A\subseteq B$ 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 - Oct 28th 2009, 02:35 PMleinadwerdna
how do you prove the reverse of that

if A X A subset B X B, then A subset of B - Oct 28th 2009, 02:48 PMPlato
- Oct 28th 2009, 03:10 PMleinadwerdna
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 - Oct 28th 2009, 03:14 PMPlato
- Oct 28th 2009, 03:26 PMleinadwerdna
thanks