SOLUTIONS SOLUTIONS

Math 2320.03
Quiz 1 Version 1     September 29, 2000

  1. Let f:N Z be defined by f(n) = n.

    1. Does f have a left inverse?
      If no, give a proof which explains why not.
      If yes, just give a formula for a left inverse of f.
      Answer: The function is one-to-one so itmust have a left inverse.
      Let g:Z N be defined by
      g(n) =

      n if n 0
      0 if n < 0
      (Note that there are many other correct choices for g.)
      Then gf = iN.

    2. Does f have a right inverse?
      If no, give a proof which explains why not.
      If yes, just give a formula for a right inverse of f.
      Answer: No. f is not onto as, for example, -1 \not f(N).

  2. Prove that N and {0,1,4,9,16,25,}, the set of squares in N, have the same cardinality by giving a formula for a bijection N {0,1,4,9,25,}.
    You do not need to prove that your answer is correct.
    Answer: f(n) = n2.
  3. Consider the following list of sets:
    ,{1,3,5,7},{1,2,3,4}, N, Q

    1. Identify three sets on the list with the same cardinality or explain why this is impossible.
      Answer: Just look!
      has cardinality 0.
      Both {1,3,5,7} and {1,2,3,4} have cardinality 4.
      Neither of NQ are finite. They do have the same cardinality 0.
    2. Find a set on the list with the same cardinality as \mathbbR, the set of real numbers, or explain briefly why none exists.
      Answer: None exists. \mathbbR is infinite as are N and Q. Both N and Q are countable but \mathbbR is not.


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On 2 Oct 2000, 20:33.