Proving Inverse Functions: g(x) \neq g(y) if x \neq y

[jgens]

Homework Statement

Suppose g is a function with the property that g(x) \neq g(y) if x \neq y. Prove that there is a function f such that f \circ g = I

Homework Equations

A function is collection of ordered pairs with the property that if (a,b) and (a,c) are in the collection, then b = c.

The Attempt at a Solution

Here are my thoughts so far (this isn't a proof):

From the restrictions on g we have that g(a) = g(b) if and only if a = b; hence, g is the collection of ordered pairs (x,g(x)) with the property that (a,c) and (b,c) are in the collection if and only if a = b.

If the composition of f and g is the identity function then f \circ g is the collection of ordered pairs ((x,g(x)),(f \circ g)(x)) = ((x,g(x)),x) = (g(x),x) such that if (a,b) and (a,c) are in the collection, then b = c. Since g has this property, it seems like there should be some function f such that f \circ g = I.

Now, I'm sure that what I've shown so far is an abuse of/improper notation and probably is not valid. I would just like some help to string everything together. Thanks!

 

[slider142]

Since you are defining a function as a collection of ordered pairs, why not define f to be the collection of ordered pairs that corresponds to reversing the order of the pairs in the collection g(x). Then the fact that fog = I is by definition, and you have only to prove that f is a function, which should be automatic given the definition of g.

 

[jgens]

I actually thought about that about a half-hour after my last post. Thanks!