Definitions and Mappings of Sets

Date: 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/