Math Is Fun Forum

xXxEmZyxXx

Member

Registered: 2007-10-15

Posts: 12

functions

let a,b be sets
suppose there is a function g:B->A with the property that g(f(a))=a for all aEA. show that  f has to be surjective

please help!!!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: functions

Hmm, let me phrase the question more clearly.

“Let A, B be sets and f:A→B be a function. Suppose there is a function g:B→A with the property that g(f(a)) = a for all a ∈ A. Show that if g is injective f has to be surjective.”

Let b ∈ B. Let c = g(b). Then c ∈ A and by the property of g, g(f(c)) = c = g(b). Since g is injective, f(c) = b; hence f is surjective.

NB: g must be injective, otherwise it won’t work. For example, if A = {x}, B = {1,2}, f(x) = 1, g(1) = g(2) = x, then f is not surjective because there is no a ∈ A such that f(a) = 2.