ONTO and INTODate: 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/ |
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