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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Information on Exam 1}
\centerline{September 20, 2012}
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\noindent
{\it General Information}
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As announced in the course syllabus and the schedule on
the course homepage, the first exam this semester will be
given Friday, September 28.  The exam will cover the
material we have discussed since the start of the semester, through
the material on matrix multiplication from class on September 21.  
The topics to review are:
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\item{1)}  Sets, set operations: union, intersection, complement, set
difference, Cartesian products and their properties
\item{2)}  Mappings (functions), the domain and range, 
direct and inverse images, injectivity and surjectivity
\item{3)}  Binary operations, the properties of associativity, commutativity,
existence of identity elements and inverses, what it means for a subset
of a set to be closed under a binary operation
\item{4)}  Composition of functions, identities and inverses for composition
\item{5)}  Matrix sums and products, and their properties.
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\noindent
{\it Practice Problems}
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The exam will consist of 4 or 5 problems (each possibly having
several parts).  Some questions will be computational, some 
will ask for definitions and/or (short) proofs.  The questions 
will be similar to parts of the following questions from Gilbert 
and Gilbert:
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\item{1)} Section 1.1: 5, 19, 23, 31, 33
\item{2)} Section 1.2: 3, 4 (be prepared to give complete proofs
why functions are 1-1 or onto, or not, in cases like this), 13, 24
\item{3)} Section 1.3: 3, 4, 5, 6
\item{4)} Section 1.4: 2jm, 3, 10
\item{5)} Section 1.5: 1de, 2, 3, 4, 5
\item{6)} Section 1.6: 2, 3, 4, 12, 22 (note: the $(a a)$ row
can be either the first or the second row of the matrix).
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\noindent
{\it Suggestions on How To Prepare}
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Read over your notes from class (several times, if necessary!). 
Make sure you can follow all the steps in proofs and examples. 
Review the problems from the problem sets, and pay particular 
attention to any problems you missed the first time.

After you do that, try some of the practice problems above. 

When you feel prepared, select four or five of the problems
above (maybe just one or two parts for the ones with lots
of parts), mixing several computational ones and some proofs.
Give yourself a practice exam using those problems, timing
yourself to make sure you can work under time pressure.
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\noindent
{\it Review Session}
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I would be willing to run an evening review session before the exam, but
scheduling it will be tricky, since I have other commitments 
Monday, Tuesday and Wednesday  
evenings.  Thursday would be OK. 
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\noindent
{\it Sample Exam}
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\noindent
{\it Disclaimer:}  This was the first exam I gave the last time I
taught MATH 243.  The exam for this class will be roughly the same 
length and the same degree of difficulty, but the questions might
be different in format and organization.
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\noindent
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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\noindent
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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\noindent
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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\noindent
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.
\bye
