Math Is Fun Forum

dchilow

Member

Registered: 2007-03-05

Posts: 27

Help with one-to-one and onto function!

I need to find a function that is one-to-one and onto between the set A:{(x,y); x,y∈Z+} another words, the set of ordered pairs (x,y) such that x and y are positive integers, and the set of positive integers.
So, I have to come up with f:A to Z+  such that f(x,y) is one-to-one and onto.

Please someone help, I have been trying to come up with one for a while.  I was trying to maybe come up with another set that I know has a one-to-one and onto function g with respect to the positive integers that would be easier to come up with a one-to-one correspondence between A and that set, then I could conclude that f o g is one-to-one and onto.  I don't know if that even makes any sense, but if not, I just need a one-to-one and onto function from A to Z+.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help with one-to-one and onto function!

What you suggested is a very good strategy for difficult mappings.  Once you get the right idea for this one however, it's not all that bad.

(1, 1), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3), (4, 4), (5, 1), ....

Typically, you can say "This mapping is defined by sending 1 to ____, 2 to ____, ..." basically saying "follow the pattern".  However, you can define it algorithmically by saying what the next step is if x = y and what it is if x is not equal to y.  If you are required to come up with an algebraic formula for it, that is going to be more difficult.

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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..."