3.2 Sets Defined by Propositions

The most common way of describing sets is by means of proposition forms.

3.24   Notation ($\{x:P(x)\}$) Let $P$ be a proposition form over a set $S$, and let $T$ be a subset of $S$. Then

$$\{x: x\in T \mbox{ and } P(x) \}$$ (3.25)

is defined to be the set of all elements $x$ in $T$ such that $P(x)$ is true. The set described in (3.25) is also written

$$\{ x \in T : P(x) \}.$$

In cases where the meaning of ``$T$'' is clear from the context, we may abbreviate (3.25) by

$$\{ x:P(x) \} .$$

3.26   Examples.

$$\{ x \in \mbox{{\bf Z}} : \mbox{ for some } y \in \mbox{{\bf Z}} ( x=2y) \}$$

is the set of all even integers, and

$$\mbox{${\mbox{{\bf Z}}}^{+}$} = \{x: x \in \mbox{{\bf Z}} \mbox{ and } x > 0 \}.$$

If $A$ and $B$ are sets, then

$$A \cup B = \{x: x \in A \mbox{ or } x \in B \},$$ (3.27)

$$A \cap B = \{x: x \in A \mbox{ and } x \in B \},$$ (3.28)

$$A \setminus B = \{x: x \in A \mbox{ and } x \not\in B \}.$$ (3.29)

We will use the following notation throughout these notes.

3.30   Notation ( $\mbox{{\bf Z}}_{\geq n}$, $\mbox{{\bf R}}_{\geq a}$) If $n$ is an integer we define

$$\mbox{{\bf Z}}_{\geq n} = \{ k \in \mbox{{\bf Z}}: k \geq n\}.$$

Thus

$$\mbox{{\bf Z}}_{\geq 1} = \mbox{${\mbox{{\bf Z}}}^{+}$} \mbo... ...} = \mbox{the set of non-negative integers} = \mbox{{\bf N}}.$$

Similarly, if $a$ is a real number, we define

$$\mbox{{\bf R}}_{\geq a} = \{ x \in \mbox{{\bf R}}: x \geq a\}.$$

3.31   Definition (Ordered pair.) If $a,b$ are objects, then the ordered pair $(a,b)$ is a new object obtained by combining $a \mbox{ and } b$. Two ordered pairs $(a,b) \mbox{ and } (c,d)$ are equal if and only if $a = c \mbox{ and } b = d.$ Similarly we may consider ordered triples. Two ordered triples $(a,b,x) \mbox{ and } (c,d,y)$ are equal if and only if $a = c \mbox{ and } b = d \mbox{ and } x = y.$ We use the same notation $(a,b)$ to represent an open interval in $\mbox{{\bf R}}$ and an ordered pair in $\mbox{{\bf R}}^2$. The context should always make it clear which meaning is intended.

3.32   Definition (Cartesian product) If $A,B$ are sets then the Cartesian product of $A$ and $B$ is defined to be the set of all ordered pairs $(x,y)$ such that $x \in A \mbox{ and } y \in B:$

$$A \times B = \{ (x,y) : x \in A \mbox{ and } y \in B \}$$ (3.33)

3.34   Examples. Let $a, b, c, d$ be real numbers with $a\leq b$ and $c\leq d$. Then

$$[a,b]\times [c,d] = B(a,b:c,d)$$

and

$$[c,d]\times [a,b] = B(c,d:a,b).$$

Thus in general $A \times B \not= B \times A$.

The set $A \times A$ is denoted by $A^2$. You are familiar with one Cartesian product. The euclidean plane $\mbox{{\bf R}}^2$ is the Cartesian product of R with itself.

3.35   Exercise. Let $S = B(-2,2:-2,2)$ and let

$R_1 = \{ (x,y) \in S : xy \leq 0 \}$

$R_2 =\{ (x,y) \in S: x^2-1 \leq 0 \}$

$R_3 = \{(x,y) \in S: y^2-1 \leq 0 \}$

$R_4 = \{(x,y) \in S : xy(x^2-1)(y^2-1) \leq 0 \}$

Sketch the sets $S,R_1,R_2,R_3,R_4$. For $R_4$ you should include an explanation of how you arrived at your answer. For the other sets no explanation is required.

3.36   Exercise. Do there exist sets $A,B$ such that $A \times B$ has exactly five elements?