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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Midterm Exam 1 -- September 29, 2006}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible points.
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I.   For this problem, 
$$\eqalign{U &= {\bf Z}\cr
           A &= \{ x \in {\bf Z} : x {\rm\ is\ odd \ and\ } -10 \le x \le -4\}\cr
           B &= \{-2,-1,0,1\}\cr
           C &= \{x \in {\bf Z} : x < 0 \}\cr}$$
and $f: {\bf Z} \to {\bf Z}$ is the mapping defined by 
$$f(x) = \cases{x+3 & if $x$ is even \cr
                x-1 & if $x$ is odd\cr}.$$
\item{A)} (10) What is the set $A \cup (B \cap C')$?
\item{B)} (10) What is the set $f(B)$?
\item{C)} (10) Given: $f$ is a permutation of ${\bf Z}$ (you don't need
to show this).  Find the inverse mapping $f^{-1} : {\bf Z} \to {\bf Z}$.  
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II.  Let $f : A \to B$ be any mapping.
\item{A)}  (15) Prove that for any subsets $T_1$ and $T_2$ of $B$,
$f^{-1}(T_1 \cup T_2) = f^{-1}(T_1) \cup f^{-1}(T_2)$.  
\item{B)}  (10) Is $f^{-1}(f(S))$ always equal to $S$ for 
all subsets $S \subseteq A$?  If you say no, give a counterexample;
if you say yes, say why.
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III.  Let $*$ be the binary operation on ${\bf R}$
defined by $x*y = x y - 2 x$. 
\item{A)} (10) Show with specific counterexamples that 
$*$ is {\it not associative}, and {\it not commutative}.
\item{B)} (15) An element $e$ is said to be a 
{\it left identity} for a binary operation if 
$e * x = x$ for all $x$.  Similarly, $e$ is a 
{\it right identity} if $x * e = x$ for all $x$.  
Does the $*$ operation have a left identity?
Does it have a right identity?  If so, say what $e$ is
in each case.  If not, explain how you reach your conclusions.  
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IV.  
In both parts of this problem, 
$$A = \pmatrix{1 & 1 \cr 3 & 4} \qquad  B = \pmatrix{4 & -1\cr -3 & 1\cr}.$$
\item{A)} (10) Compute the matrix product $AB$ or say why it doesn't exist.
\item{B)} (10) Define a mapping 
$$F : M_{2\times 2}({\bf R}) \to  M_{2\times 2}({\bf R})$$
for $X$ a $2\times 2$ matrix by by $F(X) = BX$ (matrix product).  
Prove that $F$ is one-to-one (injective) and give its left inverse mapping.
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