Theorem 5.9: Let S be any set and let {a } be a set with one element. Then #(S x {a }) = #(S ).

Proof: Define a function

by

f ((s , a )) = s

By Theorem 5.5, it suffices to show that f is a one to one correspondence. Suppose that

f {(s , a )) = f ((t , b ))

By the definition of f

s = t

and since the second coordinate has to be a,

(s , a ) = (t , b )

To show that f is onto, let

Then

s = f ((s , a ))