Surjection: Is f^-1(X) Surjective? Why?

[patfan7452]

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

 

[Hurkyl]

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

Anyways, have you looked at any examples?

 

[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?

 

[HallsofIvy]

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.