Question: > 
> For sets $A,B,C$ not equal to $\emptyset$ prove that $$AXB=BXA$$ $$\iff$$ $$A=B$$
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Answer: 
Hint: Just think about the definitions of $\cap$ and $\cup$ and the following: every element lying in $A$ and $B$ or $C$ is element of $A$ and $B$ or $A$ and $C$ (and reverse).
This is the distributive law for set operations. 

<http://www.cs.odu.edu/~toida/nerzic/level-a/set/set_op_prop.html>

This site has list of set operations and some proofs which will show you one way to do this.