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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 1 -- Sets and Functions}
\centerline{September 2, 2005}
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{\it Background}
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Today, we will review (or learn) some language about sets and
functions.  To begin, for us, a {\it set} will be a collection
of objects, called the {\it elements} of the set.  Usually, but
not always, the elements of a set will be mathematical ``things''
like numbers, geometric figures, and so forth.  Usually sets 
are named by capital letters like $A,B,X,Y$, and elements are
named $a,b,x,y$, etc.   The statement that $x$ is an element
of the set $A$ is written in symbolic form as $x \in A$
(the symbol $\in$ means ``is an element of'').  We can specify
sets by either 
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\item{$\bullet$} listing the elements,
\item{$\bullet$} giving a ``recipe'' for constructing all the elements, or
\item{$\bullet$} giving a ``test'' for whether an element of a 
larger ``universal'' set is contained in the set we are talking about.
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For instance, consider the set of natural numbers $S = \{1,3,5,7,9\}$.
We could also define the same set as 
$$S = \{2k + 1 : 0 \le k \le 4, k \in {\bf N}\}$$
or as
$$S = \{n \in {\bf N} : n {\rm\ is\ odd\ and\ } n \le 9\}$$
(among many other ways).  Today, we will look at a number
of examples.
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{\it Discussion Questions}
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A) Given sets $A,B$ inside some universal set $X$,
we can make other sets called the {\it union} $A \cup B$,
the {\it intersection} $A \cap B$, and the {\it complement}
$A^c$ (complement inside $X$).
How are each of these defined?  If you don't know 
or don't remember, look in section 1.2 of the text, 
and explain what is there in your own words.
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B) Let the universal set be $X = {\bf Z}$ (the set of integers) and let 
$$A = \{n \in {\bf Z} : n^2 < 12\} \qquad B = \{ n \in {\bf Z} : 1 < 2n + 5 < 20\}$$
\item{1)}  List the elements in $A,B$.  
\item{2)}  What are the sets $A \cap B$, $A \cup B$ and $A^c$?
\item{3)}  Show that $(A \cup B)^c$ is the same set as
$A^c \cap B^c$ by describing the elements of each.
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C)  For each natural number $n$, let $A_n = \{ x \in {\bf R} : x \ge n\}$.
(We could also write the set $A_n$ using interval notation as
$A_n = [n, +\infty)$.)  Determine the following sets:
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\item{1)}  $A_1 \cap A_2$, $A_1 \cap A_2 \cap A_3$, $A_1 \cap A_2 \cap A_3 \cap A_4$.
\item{2)}  What set is the intersection of {\it all} the $A_n$'s:
$$\bigcap_{n=1}^\infty A_n = A_1 \cap A_2 \cap A_3 \cap \cdots\ ?$$
Explain.
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D)  Let $A,B$ be sets.  A {\it function} $f : A \to B$ is a rule
or mapping that to each element $x \in A$ assigns a single 
element $f(x) \in B$.  The set $A$ is called the domain of $f$.
The set of elements $f(A) = \{y \in B : y = f(x) {\rm\ for\ some} x \in A\}$
is called the range of $f$.
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\item{1)}  Let $f : {\bf R} \to {\bf R}$ be the function defined
by $f(x) = x^2 + 1$.  What is the range of $f$?
\item{2)}  Same question for $f: [-1,1] \to {\bf R}$ defined
by $f(x) = \arctan(x)$.  (Note the domain!)
\item{3)}  How would you define the {\it graph} of a function
using set notation?  (What new type of object is involved?)
\item{4)}  Consider the function $f : {\bf R} \to {\bf R}$
defined by 
$$f(x) = \cases{x & if $x \in {\bf Q}$\cr
                0 & if $x \notin {\bf Q}$\cr}$$
Draw (or describe in words) the graph of this function.
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{\it Assignment}
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Writeups due in class on Monday, September 5.
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