Theorem 3.2

Theorem 3.2: If A, B, and C are sets and

and

are functions where f and g are both one to one, then so is gf.

Proof: By the definition of a one to one function, Definition 3.5, we must show that if

gf(a) = gf(b).

then

a = b.

Assume that

gf(a) = gf(b).

That means

g(f(a)) = g(f(b))

But since g is one to one, that means that

f(a) = f(b)

But since f is one to one, that means that

a = b.