Definitions and Mappings of SetsDate: 09/22/98 at 03:26:32 From: Bryan Wang Subject: Basic topology Dear Dr. Math: I am studying mathematical analysis out of Rudin and have the following questions: (1) The definition of a 1-1 mapping is: If for each y in B, f^(-1)(y) consists of at most one element of A, then f is said to be a 1-1 mapping of A into B, where A and B are sets. Consider the mapping g of A into B. If g(x1) = g(x2) for all x1, x2 in B and x in A with x1 = x2, then is f' a 1-1 mapping of A onto B? According the definition, I think it is. What do you think? (2) If g, in (1), is a 1-1 mapping of A onto B, then how does the relation of the symmetric property B~A hold? Since f'^-1 is not a 1-1 mapping of B onto A. (3) If a set A is finite, then is it countable? (4) If a set A is infinite, then is it countable? Thank you. Bryan Wang Date: 09/22/98 at 08:25:55 From: Doctor Jerry Subject: Re: Basic topology Hi Bryan, (1) In your statement, you are saying that a function g from A into B is 1-1 provided that for all x and y in A, if g(x) = g(y), then x = y. Yes, this statement is sometimes used as a definition. It is equivalent to the statement that f is 1-1 if for all y in B, f^{-1}(y) is a subset of A and has at most one element. (2) Rudin, the author of the text, is using f^{-1} in two different ways. Given a function f from A to B and any subset H of B, f^{-1}(H) is the set of all a in A for which f(a) is in H. If H is a set with one element, then f^{-1}(H) is a SET with one element, something like {q}. Rudin, page 90, defines a slightly different f^{-1}, when f is 1-1 onto B. Basically, he defines f^{-1}(q) = p, where f(p) = q. (3) Rudin uses slightly different terminology on countable sets than most books. Most mathematicians would say that all finite sets are countable. Rudin says that all finite sets are at most countable. So, for Rudin, if a set is finite, then it is not countable. Notice Theorem 2.8. Rudin says Every infinite subset of a countable set is countable. (4) No, this is false. Theorem 2.14 gives an example of an infinite, non-countable set. The set of rational numbers is countable, but the set of all reals is infinite and not countable. Cantor gave a proof of this, using his famous diagonal construction. This construction is in the proof of Theorem 2.14, but the context is different than Cantor's construction. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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