Theorem 3.5 (The Associative Property of Composition of Functions): If A, B, C and D are sets and

and

are functions then

(hg )f = h (gf )

Proof: Let a be an element of A. Then

((hg)f)(a) = hg(f(a)) = h(g(f(a)))

and

(h(gf))(a) = h(gf(a)) = h(g(f(a)))