ONTO and INTO

Date: 07/27/2001 at 10:29:57
From: Pawntep Shaikitwattana
Subject: The difference between ONTO and INTO when you describe a 
function

Dear Sir,

I've always been confused with ONTO and INTO.
I learned a chapter about functions when I was in high school and 
proceeded to higher mathematics without deep knowledge on functions.
If you can give me a simple explanation, I would greatly appreciate 
your help.

Thank you very much.

Best regards,
Pawntep

Date: 07/27/2001 at 12:09:00
From: Doctor Peterson
Subject: Re: The difference between ONTO and INTO when you describe a 
function

Dear Pawntep:

A function takes points in a domain and moves them to points of the 
range. Let's consider a function f from set A to set B.

An "onto" function, also called a "surjection" (which is French for 
"throwing onto") moves the domain A ONTO B; that is, it completely 
covers B, so that all of B is the range of the function. Formally, for 
every point y in B, there is some point x in A such that f(x) = y. 
Informally, imagine A as a blanket that is thrown so as to completely 
cover a bed; there is no point on the bed that does not lie under the 
blanket.

An "into" or "one-to-one" function, also called an "injection" (which, 
as you can guess, is French for "throwing into") moves the domain A 
INTO B, in the sense that each point in A retains its identity as a 
distinct point in B. That is, no more than one point in A is moved to 
any given point in B; there are no two points x1 and x2 in A such that 
f(x1) = f(x2). This makes it possible to invert the function, since 
for every point in the range there is a unique point in A to go back 
to. Whether or not the blanket we threw over the bed completely 
covered it, we have thrown it "into" the bed if there are no wrinkles, 
so that no point on the bed has more than one layer on it. You may 
imagine that the bed melts the blanket, so that any wrinkles would be 
stuck together and the blanket would no longer be the same blanket you 
started with. That's a poor analogy, but I think it gets the point 
across.

You can read about this here:

   Injective, Surjective, Bijective Functions
   http://mathforum.org/dr.math/problems/david.01.23.01.html   

(You will have to replace the phrase "at most" with "at least" to make 
sense of the definition given here. It is corrected in the answer, but 
never actually said to be wrong.) I'm not sure why you didn't find 
this by searching for "onto function"; but you will not find "into," 
both because the searcher ignores it and because the common term in 
English is "one-to-one."

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/