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\centerline{MATH 243 -- Mathematical Structures}
\centerline{Quiz 8 -- November 17, 2017}
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\noindent
Your Name: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
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\noindent
{\it Directions}
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Do all work and answer all questions on this sheet. (Use the back if you need more space.)  There are 30 possible points, 
distributed as indicated.    
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\begin{enumerate}
\item[A)]  Let $f : \mathbb{Z} \to \mathbb{Z}/12\mathbb{Z}$ be the mapping defined by 
$f(x) = [x + 3] \in \mathbb{Z}/12\mathbb{Z}$.  
\begin{enumerate}
\item[(1)] (7) Is $f$ injective?  Why or why not?
\vglue .5truein
\item[(2)] (7) Let $T = \{x \in \mathbb{Z} : x {\rm\ is\ odd}\} \subset \mathbb{Z}$. 
What is $f(T)$?  (Describe using correct set notation; you need not give a full proof of your assertion.)
\vglue 1.5truein
\item[(3)] (6) Let $U = \{[2],[4]\} \subset \mathbb{Z}/12\mathbb{Z}$. What is $f^{-1}(U)$?
(Describe using correct set notation; you need not give a full proof.)
\vglue 1.5truein
\end{enumerate}
\item[B)] (10) Prove that if $f : A \to B$ is a mapping, and $b_1 \ne b_2$ are any two distinct
elements of $B$, then $f^{-1}(\{b_1\}) \cap f^{-1}(\{b_2\}) = \emptyset$.   ({\bf Note}:  This 
is a completely general statement.  You may not use properties of any particular example, 
only the definitions of the concepts involved here.)
\end{enumerate}
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