1. Short Answer.
    1. Find the smallest counterexample to the statement: MATHMATH
      MATH

      MATH
    2. Let MATHand MATHFind $left( iright) $$gcirc f$if it exists and $left( iiright) $$fcirc g$if it exists.
      MATH
    3. Let MATH. Write the contrapositive of the following statement: If $x$and $y$are both positive then $dfrac{x}{y}leq 1$or MATH
      MATH
  2. Prove the following by either the Method of Smallest Counterexample or by the Method of Mathematical Induction.
    MATH


    3. MATH

  3. Prove the following by the Method of Proof by Contradiction: If $A,B,$and $C$are sets such that $A-B=emptyset $then MATH
    MATH
  4. Let $A$be a set. Let $f:Arightarrow A.$Prove: If $f$is onto then $fcirc f$is onto.
    MATH
  5. Let MATHbe given by $f(x)=2x-1.$Determine if $f$is $left( iright) $one-to-one and $left( iiright) $onto. Prove your assertions. (That is, either give a proof if the property does hold or a counterexample if it does not.)
    MATH

    MATH
  6. Either give an example that satisfies the stated conditions or explain why it is impossible to do so. $ $(You may give your examples in the form of a function diagram, as long as everything in the diagram is clear and fully labeled.)
    1. A function MATHsuch that $f$is one-to-one, but not onto.
      MATH
    2. A pair of sets $A$and $B$and a function $f:Arightarrow B$that is onto but not one-to-one.
      MATH
    3. Sets $A,B,$and $C$and functions $f:Arightarrow B$and $g:Brightarrow C$such that $f$is one-to-one but $gcirc f$is not one-to-one.
      MATH
    4. A function MATHsuch that $f^{-1}$is not a function from $QTR{Bbb}{Z}$to $QTR{Bbb}{Z}.$
      MATH
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