Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Showing a function is one-to-one and onto.

Let:

To show f is one-to-one:

Assume:

and we need to show that:

So we get:

How do we go from here?

Last edited by LuisRodg (2008-11-21 01:20:10)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Showing a function is one-to-one and onto.

i and j must be integers.  This means that you should be using the properties of integers, not polynomials (where techniques are dedicated to real numbers), to solve the problem.  What happens if i is even?  What about i odd?  And other cases that you can think of.

The above may not help at all in solving the problem, but this is how you investigate mathematical structure.  Start with a special case, and see what holds.  If you can come up with a finite numbers of special cases that cover all possible cases (odd and even, prime and not prime, 13 and not 13, etc), then you've solved the problem.  Even if you can't, it typically gives insight into what is really going on, something you can't see from a formula.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."