Expert: Sherry Wallin - 6/7/2010

Question
Hi!sherry wallin sir.
I am always confusion about the one to one crosspoundece function and what is the physical meaning of the this function.
             
                                               your sincerly
                                                 hari har kafle

Answer
Hi Hari Har~
    I think what you meant is what is a one-to-one correspondence. A one-to-one correspondence is also known as a bijection which is just to say that the function is both  one-to-one and onto or injective as well as surjective. Think of an ordered pair on your x-y coordinate plane and say you have three ordered pairs (2,4) (1,1) (0,0). If there is a rule that generates those ordered pairs (and I will make one up) say f(x) = x^2, then for every x there is just one y value that would work which makes it a function and furthermore no two x's are associated with the same y value. So a function is injective if different x's are always paired with a different y. In our example above 2 is mapped to 4, 1 is mapped to 1, and 0 is mapped to 0. You see, the x's: 0,1,4 are each mapped to a different y"s: 0,1,2 this makes the function one-to-one (injective).
And as long as the range is the same size as the domain (there are the same number of x's as y's and every y is paired with any x then the function is onto. And if by chance both occur (one-to-one and onto) then you have a 1-1 correspondence. In other words you can picture the x's in a circle and the y's in a circle and if you chose an x there would be a y that would be paired with it and only one y that would be paired with that x.

But I suspect what you really want is an example of how this might be used. Relational databases depend heavily on injective, surjective, and bijective functions. When you query a database to say find someone's fingerprints you are pretty certain that there is only one set that would match a certain person. So an association is used to tie this relationship together so that when those fingerprints are put in again, they always return the same person. But what if that person has many aliases and the fingerprints are associated with each of those aliases? Then we know that all those aliases are the same person. So pretend the the names of people are in the domain and the fingerprints are in the range. When we input the person's name we will get a match with a fingerprint. And let's just say that all we have in this domain is all the aliases for that one fingerprint in the range. So every alias is associated with that one fingerprint. In order for this to be one-to-one that would have to be that all those (seemingly different) aliases are one and the same, since different elements in the domain are mapped to different elements in the range and all these aliases would get mapped to the one fingerprint in the range thus if we make the function one-to-one we could logically deduce that all those aliases are one and the same person. Furthermore since the range is just that one set of fingerprints then every element in the range was associated with an element from the domain and thus is onto. Since the function is both onto and one-to-one we have a one-to-one correspondence between the aliases and the fingerprints.

I hope this has been helpful, it took quite a bit for me to think of a specific example that works.

Math Prof