The intersection of two sets $A$ and $B$, denoted by $A \cap B$, consists of all elements common to both sets. In set-builder notation:

$A \cap B = \{ x : x \in A \text{ and } x \in B \}$

Examples

1. If $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and $B = \{1, 3, 5, 7, 9\}$, then: $A \cap B = \{1, 3, 5, 7\}$
2. If $A = \{a, e, i, o, u\}$ and $B = \{a, b, c, d, e, f, g, h, i\}$, then: $A \cap B = \{a, e, i\}$ (joint sets)
3. If $A = \{a, e, i, o\}$ and $B = \{b, c, f, g\}$, then: $A \cap B = \emptyset$ (disjoint sets)

The complement of a set $A$, denoted by $A'$ or $A^c$, includes all elements in the universal set $U$ that are not in $A$:

$A' = \{ x \in U : x \notin A \}$

Examples

1. If $U = \{a, b, c, ..., z\}$ and $A = \{a, b, c, d, e\}$, then: $A' = \{f, g, h, ..., z\}$
2. $U = \{125, 245, 365, 475, 585\},\ A = \{365\} \Rightarrow A' = \{125, 245, 475, 585\}$
3. $U = \{a, e, i, o, u\},\ B = \{e, i, u\} \Rightarrow B' = \{a, o\}$

A number line can be used to represent a set of values in a given interval. The number of elements within the interval cannot be counted or listed because they are infinite. An interval on the number line can be open, closed, half-open, or half-closed.

• Open interval: An open interval excludes the endpoints. For example, if $a < b$, the open interval from $a$ to $b$ is denoted by $(a, b)$ or $\{ x : a < x < b \}$.
• Closed interval: A closed interval includes both endpoints. It is denoted by $[a, b]$ or $\{ x : a \leq x \leq b \}$.
• Half-open/half-closed intervals:
  • Left-closed, right-open: $[a, b)$ or $\{ x : a \leq x < b \}$
  • Left-open, right-closed: $(a, b]$ or $\{ x : a < x \leq b \}$

Table: Forms of intervals on a number line

Interval Notation	Number Line Representation	Meaning
$(a, b)$	Left open, right open	$a < x < b$
$[a, b]$	Left closed, right closed	$a \leq x \leq b$
$(a, b]$	Left open, right closed	$a < x \leq b$
$[a, b)$	Left closed, right open	$a \leq x < b$
$(-\infty, a]$	Right closed	$x \leq a$
$(-\infty, a)$	Right open	$x < a$
$(a, \infty)$	Left open	$x > a$
$[a, \infty)$	Left closed	$x \geq a$

Example

Find $A \cap B$ given:

$A = \{ x : x \geq -2 \}, \quad B = \{ x : x < 5 \} \text{ where } x \in \mathbb{R}$

Solution:

$A \cap B = \{ x : -2 \leq x < 5 \}$

Example

Find $A \cup B$ given:

$A = \{ x : x \geq -1 \}, \quad B = \{ x : x \leq 6 \} \text{ where } x \in \mathbb{R}$

Solution:

$A \cup B = \{ x : x \in \mathbb{R} \}$

Example

Find $A - B$ given:

$A = \{ x : x > 3 \}, \quad B = \{ x : x < 5 \} \text{ where } x \in \mathbb{R}$

Solution:

$A - B = \{ x : x \geq 5 \}$