Providing counterexamples to the claim: If $|A \cap B| < |A|$ then $|A|>|B|$

Question

I'm having problems answering the following question:

Give a counter example to show that the following statement is false:

For any sets $A$ and $B$, if $|A \cap B| < |A|$ then $|A|>|B|$

Presumably we can't be sure whether $|A|>|B|$ as we don't know how many elements there are in $B$ which aren't shared with $A.$

Answer 1

You're on the right track. See what happens when $A$ and $B$ have no elements in common at all. Can you build a counterexample in this case?

Answer 2

You have the right idea. So you should be able to construct your counterexample if you start with some sets that already satisfy $|A\cap B|<|A|$, and then add enough fresh elements to $B$ to make $|A|\not>|B|$.