Question: If g is one-to-one and gf is onto, prove that f is onto. Given: Let X,

Question

Question: If g is one-to-one and gf is onto, prove that f is onto.

Given: Let X,Y,Z be subsets of U, the universal set. Let f be from X to Y. Let g be from Y to Z. Let

(g o f)(x) = g(f(x)) be the definition of composition.

Attempt at solution: Let x,y,z be in X,Y,Z respectively. Assume that for all z in Z, there exists a x in X such that (g o f)(x) = g(f(x)) = z. Also assume that for all y,y' in Y, g(y) = g(y') implies that y = y'.

That's pretty much as far as I've gotten. I do know that if (g o f), or gf, is onto that this implies g is also onto and can use this fact in my proof. I am really stuck and have been at this for hours. Any help is really appreciated and thank you in advance!

Explanation / Answer

Let y be in Y. We wish to find an x such that f(x) = y. g(y) is in Z. Since gf is onto, there exists an x such that g(f(x)) = g(y). This x is precisely the x that we need. Since g is one-to-one and g(f(x)) = g(y), we must have f(x) = y. Thus, f is onto, as desired.

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.