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\centerline{Mathematics 243 -- Algebraic Structures}
\centerline{Discussion 1 -- Binary Operations}
\centerline{September 13, 2006}
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\noindent
{\it Background}
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Last time we introduced the idea of a {\it binary operation}
on a set $A$, that is a mapping:
$$\eqalign{
* : A \times A &\to A\cr
    (a,a') &\mapsto a*a' \cr}$$
We said
\item{$\bullet$}  $*$ is {\it associative} if (and only if)
$(a*a')*a'' = a*(a'*a'')$
for all $a,a',a'' \in A$.
\item{$\bullet$}  $*$ is {\it commutative} if (and only if)
$a*a' = a'*a$
for all $a,a' \in A$.
\item{$\bullet$}  An element $e \in A$ is an {\it identity 
element for $*$} if 
$e * a = a \quad {\rm\ and\ } \quad a * e = a $
for all $a\in A$.  

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One additional concept:
\item{$\bullet$}  If $*$ is a binary operation on $A$ that
has an identity element $e$ (this is necessary for what follows), then
we say $b \in A$ is an {\it inverse} of $a$ if 
$a * b = e$ and $b * a = e$.  
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\noindent
Today, we want to practice with several additional examples
of binary operations and study these concepts in more detail.
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{\it Discussion Questions}
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A)  Let $* : {\bf Z} \times {\bf Z} \to {\bf Z}$ be the binary operation
defined for $x,y \in {\bf Z}$ by 
$$x * y = x + y - xy.$$
Show (i.e. prove) the following:
\item{1)}  $*$ is commutative 
\item{2)}  $*$ is associative
\item{3)}  $*$ has an identity element (you'll need to find one, then
show that it ``works'' in the definition).
\item{4)}  There are only two elements of ${\bf Z}$ that have inverses
for this operation (you'll need to find each of them and their inverses, 
say why they ``work'', then show that your elements are the only ones).
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\noindent
B)  Let $A$ be the set $\{a,b,c,d,f\}$ and consider the binary operation
defined by the following table:
$$\matrix{ * & a & b & c & d & f\cr
           a & f & d & c & b & a\cr
           b & d & c & b & a & f\cr
           c & c & b & a & f & d\cr
           d & b & a & f & d & c\cr
           f & a & f & d & c & b\cr}$$
\item{1)}  Is the operation $*$ commutative?  Why or why not?
\item{2)}  Does the operation $*$ have an identity element?  Why or why not?
\item{3)}  Is the operation $*$ associative.  Why or why not?
\item{4)}  What is true about $x^3 = x*x*x$ for all $x \in A$? 
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\noindent
C) Let $*$ be a binary operation on a set $A$.  Suppose $e_1$
is an identity element for $*$.   Can there be {\it another}
element $e_2 \ne e_1$ which is {\it also} an identity element 
for $*$.  Prove your assertion.
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\noindent
{\it Assignment}
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Writeups (one per group) due Monday, September 18. 
\bye
