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

 

 

 

 

    1. Let MATHand MATHFind $left( iright) $$gcirc f$ if it exists and $left( iiright) $$fcirc g$ if it exists.

 

 

 

 

    1. Let MATH. Write the contrapositive of the following statement: If $x$and $y$are both positive then $dfrac{x}{y}leq 1$or MATH

 

 

 

 

  1. Prove the following by either the Method of Smallest Counterexample or by the Method of Mathematical Induction.

 

 

 

 

 

 

 

 

 

 

 

  1. Prove the following by the Method of Proof by Contradiction: If $A,B,$and $C$are sets such that $A-B=emptyset $then MATH

 

 

 

 

 

 

  1. Let $A$ be a set. Let $f:Arightarrow A.$Prove: If $f$is onto then $fcirc f$ is onto.

 

 

 

 

 

 

 

  1. 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.)

 

 

 

 

 

 

 

 

  1. 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.

 

 

 

 

    1. A pair of sets $A$and $B$and a function $f:Arightarrow B$that is onto but not one-to-one.

 

 

 

 

    1. 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.

 

 

 

 

    1. A function MATHsuch that $f^{-1}$is not a function from $QTR{Bbb}{Z}$to $QTR{Bbb}{Z}.$