Suppose A, B, C are sets and consider the function?

(or mappings) f: A -%26gt; B and g: B -%26gt; C.


i. show that if f and g are both injective then so is their composition g*f


ii. show that if f and g are surjective then so is their composition g*f


recall that (g*f)(a)=g(f(a))

Suppose A, B, C are sets and consider the function?
(i) if f and g are injective then f(a) = f(b) iff a=b and g(c) = g(d) iff c=d


hence g(f(a)) = g(f(b)) iff f(a) = f(b) iff a=b





so g*f is injective





(ii) f and g are surjective iff for all b in B there exists a in A such that f(a)=b and similarly the exists b in B such that g(b) = c for all c in C


so for all c in C there exists b in B such that g(b)=c, and for all b in B there exists a in A such that f(a) = b


therefore, for all c in C there exists a in A such that g(f(a)) = c, and g*f is surjective





and as g*f is both surjective and injective it is also bijective!