I want to show if
|AxB|=|A| what can we conclude about A and B, and justify it.
I have that
|AxB|=|A|
|A|•|B|=|A|
|B|=|A|/|A|
|B|=1
so the cardinality of B is 1
Once again, have you failed to state important bits of information?
Do you know that $\displaystyle A\ne \emptyset~?$
If not, you need to discuses that possibility.