Suppose that A and B are finite sets Show that ABAB if and o

Suppose that A and B are finite sets. Show that |AB|=|A|+|B| if and only if AB =

Solution

let us assume that there is a k N {0} and a bijection f such that f : A [k].

Also there is a j N {0} and a bijection g such that g : B [j].

We name a new bijection h : A B [k + j].

We describe h as follows: h(x) = ½ f(x), x A k + g(x), x B Verify that $h$ is a bijection.

Thus |A B| = k + j = |A| + |B|.