Objective 1 - Set Notation

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Describe solutions as sets of numbers.

Link to section in online textbook.

Note: As this section reviews definitions, there is no associated video.

We start with a terminology review.

A set is a collection of mathematical objects. We’ll commonly look at sets of numbers like the Natural numbers: $\{ 1, 2, 3, 4, \ldots \}$ .

An interval is a collection of Real numbers. For example, $(1, 2)$ is the set of Real numbers between 1 and 2 (but not including 1 or 2). If we want to include the endpoints of an interval, we use brackets, such as $[1, 2]$ .

We can describe solutions that exist in an interval by using the notation $x \in (a, b)$ . We read this as “ $x$ is an element of $(a, b)$ ” and means that $x$ is some number between $a$ and $b$ . For example, $x \in [1, 2]$ means that $x$ is some number between 1 and 2 (and could be one of the two numbers).

We can also describe intervals using inequalities. For example, to describe the set of $x \in (1, 2)$ using inequalities, we would use $1 < x < 2$ . This is usually a more natural way for students to read “ $x$ is a Real number between 1 and 2.” If we want to include the endpoints of an inequality, we use the symbols $\leq$ or $\geq$ .

Write each set described in Interval notation.


Set described in words	Inequality Notation	Interval Notation

All Real numbers between $a$ and $b$ , but not including $a$ or $b$	$a < x < b$	$\{x \, \| \, x \in \answer {(a, b)} \}$
All Real numbers greater than $a$ , but not including $a$	$x > a$	$\{x \, \| \, x \in \answer {(a, \infty )} \}$
All Real numbers less than $b$ , but not including $b$	$x < b$	$\{x \, \| \, x \in \answer {(-\infty , b)} \}$
All Real numbers greater than $a$ , including $a$	$x \geq a$	$\{x \, \| \, x \in \answer {[a, \infty )} \}$
All Real numbers less than $b$ , including $b$	$x \leq b$	$\{x \, \| \, x \in \answer {(-\infty , b]} \}$
All Real numbers between $a$ and $b$ , including $a$	$a \leq x < b$	$\{x \, \| \, x \in \answer {[a, b)} \}$
All Real numbers between $a$ and $b$ , including $b$	$a < x \leq b$	$\{x \, \| \, x \in \answer {(a, b]} \}$
All Real numbers between $a$ and $b$ , including $a$ and $b$	$a \leq x \leq b$	$\{x \, \| \, x \in \answer {[a, b]} \}$
All Real numbers less than $a$ or greater than $b$	$x < a \text { or } x > b$	$\{x \, \| \, x \in \answer {(-\infty , a)} \cup \answer {(b, \infty )} \}$
All Real numbers	$x \geq a \text { or } x < a$	$\{x \, \| \, x \in \answer {(-\infty , \infty )} \}$

On exams, you will answer questions primarily using interval notation. Solve the linear equation below and choose the interval that contains the solution.

$x + 3 = 5.5$

$x = a, \text { where } a \in [-2, -1]$ $x = a, \text { where } a \in [-1, 0]$ $x = a, \text { where } a \in [0, 1]$ $x = a, \text { where } a \in [1, 2]$ $x = a, \text { where } a \in [2, 3]$