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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Exam 1 -- September 28, 2012}
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{\it Directions}
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Do all the problems below in the blue exam booklet. 
Do not place anything you want to have considered for 
credit on the exam sheet.  There are 100 regular points and 10 Extra
Credit points. 
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I.  Let $A, B, C$ be any subsets of a universal set $U$.
\item{A)}  (15) Show that if $A \subseteq B$, then $A \cap C \subseteq B \cap C$.
\item{B)}  (10)  Is it true that $A - B = A - C$ implies $B = C$?  If you think 
the statement is true, illustrate with a Venn diagram.  If you think the statement
is false, give a counterexample. 
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II. (20)  Consider the following statement:
``If $f : A \to B$ is an onto mapping with $A,B$ nonempty, then for all $b\in B$, the
set $f^{-1}(\{b\})$ is nonempty.''
Give the contrapositive form of this statement.
Are the given statement and its contrapositive true or false?  Explain.
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III. 
Let $f,g : {\bf Z} \to {\bf Z}$ be the mappings given by
$$f(x) = \cases{2x - 1 & if $x$ is even\cr
             x + 1 & if $x$ is odd\cr}
  \qquad
  g(x) = \cases{x & if $x$ is even\cr
             x + 1 & if $x$ is odd\cr}$$
\item{A)} (15) Let $S = \{1,2,3\}$ and 
$T = \{0,3,6\}$.  What is $g^{-1}(T) \cap f(S)$?
\item{B)} (10) What is the mapping $f\circ g$?  
%\item{C)} (5) Is $f$ a permutation of ${\bf Z}$?  Why or why not?
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IV.  Consider the binary operation on ${\bf Z}$ defined by $x * y = x + xy + y$.
\item{A)} (10)  Is $*$ associative?   
\item{B)} (5)  There is an identity element for $*$.  Find it and justify your claim. 
%\item{C)} (5)  Which $x \in {\bf Z}$ have inverses for $*$?
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V. (15) Let $A \in M_{2\times 2}({\bf Z})$ be the matrix $A = (a_{ij})$ defined by $a_{ij} = i-j$,
and let $B \in M_{3\times 2}({\bf Z})$ be the matrix $B = (b_{ij})$ defined by $b_{ij} = i + 2j$.
Which of the two products $AB, BA$ are defined?  Compute the one(s) that are defined.
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{\it Extra Credit} (10)   Which $x \in {\bf Z}$ have inverses for the binary operation $*$
from question IV?
%Suppose 3 people are selected by drawing lots from a group 
%of 20 for a committee.  In how many ways can the selection be made if there is 
%no difference between the committee members?  Does your answer change if
%the committee members are to be assigned to three different positions 
%(chair, secretary, at-large member)?  
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