Math 5345     Fall semester 2000

Exercises due Wednesday, September 20

Solutions are linked here.

Notation:

• = {non-negative integers}
• = {all integers}
• = {rational numbers}
• = {real numbers}

1. Consider the following statement, about a real number:
    
                   If  x  is rational, then  x² 2.
    
   1. What is the converse?
   2. What is the contrapositive?
   3. Is the converse true?   Give a proof or a counterexample, as appropriate.
   4. Is the contrapositive true?   Give a proof or a counterexample, as appropriate.
   
   (When giving a proof, you may cite facts which were proved in class.)
    
2. Find a 1-to-1 correspondence from   to  .
   Note: In class (and in the text) we gave a 1-to-1 correspondence between the positiveintegers and  .
    
3. ...
   1. Find a 1-to-1 correspondence from   to  ²,  or explain how one can be constructed.
      Here, ²  denotes the set of all ordered pairs  (a,b),  where  a and  b are rational numbers.
      Thus, we can think of  ²   as the set of points in the plane both of whose coordinates are rational numbers.
   2. Does there exist a 1-to-1 correspondence from   to  ² ?
      Prove that your answer is correct.
       
4. ...
   • Version 1 Show that there is a 1-to-1 correspondence between the half-open interval  [0,1)  and the open interval  (0,1). 
     (You may cite any facts from the text or from class.)
      
   • Version 2 Find an explicit 1-to-1 correspondence from the half-open interval  [0,1)   to the open interval  (0,1). 
     One suggestion (but you don't haveto do it this way ... ): One can study the proof of the Bernstein-Schroeder theorem (perhaps more correctly called the Cantor-Bernstein-Schroeder theorem ...), to see how it applies to specific 1-to-1 [but not onto] mappings  f: [0,1) --> (0,1)  and  g: (0,1) --> [0,1). (There are lots of reasonable possibilities for  f and  g, based on functions studied in calculus or lower level courses.) Once you figure this out, you may be able to see a fairly reasonable possibility for the desired 1-to-1 correspondence, and it might not even be necessary to mention Bernstein-Schroeder in your final writeup.
      
   If you do Version 1correctly, you will receive full credit for this problem.
   If you do Version 2correctly, you will receive full credit for this problem, and for 2 extra problems.

Comments and questions to:  roberts@math.umn.edu

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