Surjection Question

1. Nov 28, 2007

patfan7452

If a function f: N-->X is surjective , is f^-1(X) (its inverse image) also surjective? If so, why?

2. Nov 29, 2007

Hurkyl

Staff Emeritus
f^-1(X) isn't a function...

Anyways, have you looked at any examples?

3. Nov 29, 2007

Chris Hillman

Hurkyl meant: isn't a function unless f is one-one, i.e. injective and surjective, i.e. bijective. Was this a trick question from some problem set?

4. Nov 29, 2007

HallsofIvy

Staff Emeritus
No, Chris, I don't believe that's what Hurkyl meant! I started to interpret f-1(X) as if it were f-1(x) and say "that's not necessarily a function", but f-1(X) is the "inverse image" of X. It's not a function for the very good reason that f-1(X) is a set of natural numbers.

I suspect that the correct question was "If f: N->X is surjective is f-1(X)= N?" If I understand what is meant by "f:N->X", then "surjective" is irrelevant. For ANY function f:N->X, that is, "to every point in N assigns a point in X", f-1(X)= N.