prove $|A\times B|=|A|\times|B|$ using induction

Question

Prove that |A × B| = |A| × |B| for finite sets A, B.

What should be the base case for this?

ALso, for the inductive step, if I take |A|=m and |B|=n, then should I take |A|=m+1 and |B|=n+1 and prove it using induction?

Answer 1

Prove this by induction on $n$:

If $|A|=n$ then for all finite sets $B$, we have $|A\times B|=n\cdot |B|$.

And the base case is of course for $n=0$.