1.4 More Sets

1.44   Definition (Proposition Form.) Let $S$ be a set. A proposition form $P$ on $S$ is a rule that assigns to each element $x$ of $S$ a unique proposition, denoted by $P(x)$.

1.45   Examples. Let

$$P(n)=``n^2-6n+8=0'' \mbox{ for all } n\in\mbox{{\bf Z}}.$$

Then $P$ is a proposition form on $\mbox{{\bf Z}}$. $P(0)$ is false, and $P(2)$ is true. Note that $P$ is neither true nor false. A proposition form is not a proposition.

Let

$$Q(n)=``n^2-4=(n-2)(n+2)'' \mbox{ for all } n\in\mbox{{\bf Z}}.$$ (1.46)

Then $Q$ is a propostion form, and $Q(n)$ is true for all $n\in\mbox{{\bf Z}}$. Note that $Q$ is not a proposition, but if

$$R=``n^2-4=(n-2)(n+2) \mbox{ for all } n\in\mbox{{\bf Z}}''$$ (1.47)

then $R$ is a proposition and $R$ is true. Make sure that you see the difference between the right sides of (1.46) and (1.47). The placement of the quotation marks is crucial. When I define a proposition I will often enclose it in quotation marks, to prevent ambiguity. Without the quotation marks, I would not be able to distinguish between the right sides of (1.46) and (1.47). If I see a statement like

$$P(n)=n^2-6n+8=0$$

without quotation marks, I immediately think this is a statement of the form $x=y=z$ and conclude that $P(n)=0$.

1.48   Notation. Let $S$ be a set, and let $P$ be a proposition form on $S$. Then

$$\{x\in S\colon P(x)\}$$ (1.49)

denotes the set of all objects $x$ in $S$ such that $P(x)$ is true. (Read (1.49) as ``the set of all $x$ in $S$ such that $P(x)$".)

1.50   Examples.

$$\{x\in\mbox{{\bf N}}\colon x^2-6x+8=0\}=\{2,4\}$$

$$\{x\in\mbox{{\bf Z}}\colon x=2n \mbox{ for some } n\in\mbox{{\bf Z}}\}=\mbox{ set of even integers. }$$

Variations on this notation are common. For example,

$$\{n^2+n\colon n\in\mbox{{\bf Z}}\}$$

represents the set of all numbers of the form $n^2+n$ where $n\in\mbox{{\bf Z}}$.

1.51   Definition (Union, intersection, difference.) Let $A$ be a set, let ${\cal S}$ be the set of all subsets of $A$, and let $R,T$ be elements of ${\cal S}$. We define the intersection $R\cap T$ of $R$ and $T$ by

$$R\cap T=\{x\in A\colon x\in R \mbox{ and } x\in T\};$$

we define the union $R\cup T$ of $R$ and $T$ by

$$R\cup T=\{x\in A\colon x\in R \mbox{ or } x\in T\};$$

and we define the difference $R\setminus T$ by

$$R\setminus T=\{x\in R\colon x\notin T\}.$$

1.52   Examples. If $R=\{1,2,3\}$ and $T=\{2,3,4,5\}$, then

$$\begin{eqnarray*} R\cap T&=&\{2,3\} \\ R\cup T&=&\{1,2,3,4,5\} \\ R\setminus T&=&\{1\} \\ T\setminus R&=&\{4,5\}. \end{eqnarray*}$$

1.53   Definition (Ordered pairs and triples.) Let $a,b,c$ be objects (not necessarily all different). The ordered pair $(a,b)$ is a set-like combination of $a$ and $b$ into a single object, in which $a$ is designated as the first element and $b$ is designated as the second element. The ordered triple $(a,b,c)$ is a similar construction having $a$ for its first element, $b$ for its second element and $c$ for its third element. Two ordered pairs (triples) are equal if and only if they have the same first elements, the same second elements, (and the same third elements). Thus

$$\begin{eqnarray*} (a,b)=(b,a) &\mbox{$\Longleftrightarrow$}& b=a. \\ (a,b)=(x,y... ...ngleftrightarrow$}& (a=x) \mbox{ and } (b=y) \mbox{ and } (c=z). \end{eqnarray*}$$

1.54   Warning. Ordered pairs should not be confused with sets.

$$\begin{eqnarray*} \{1,2\}&=&\{2,1\}. \\ (1,2)&\not=&(2,1). \end{eqnarray*}$$

1.55   Definition (Cartesian product, $\times$.) If $A$ and $B$ are sets, we define the set $A\times B$ by

$$\begin{eqnarray*} A\times B&=&\mbox{ the set of all ordered pairs } (a,b) \mbox{... ...rdered triples } (a,b,c) \mbox{ where } a,b,c \mbox{ are in } A. \end{eqnarray*}$$

$A\times B$ is called the Cartesian Product of $A$ and $B$.

1.56   Example. If $\mbox{{\bf R}}$ is the set of real numbers, then $\mbox{{\bf R}}^2$ is the set of all ordered pairs of real numbers. You are familiar with the fact that ordered pairs of real numbers can be represented as points in the plane, so you can think of $\mbox{{\bf R}}^2$ or $\mbox{{\bf R}}\times\mbox{{\bf R}}$ as being the points in the plane.