Set-builder Notation

5 min. | Beginner | (Hammack 2013)

Set-builder Notation

A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces. Leveraging set-builder notation a particular set \mathcal{X} could be described as:

\mathcal{X} = \{ \text{expression}: \text{rule} \};

where the elements of \mathcal{X} are values of an expression following a particular rule.

Set-Builder Notation: Example

The set \mathcal{A}_{\text{set notation}} for all even integers:

\mathcal{A}_{\text{set notation}} = \{..., -4, -2, 0, 2, 4, ...\}

This set can be described using set-builder notation:

\mathcal{A}_{\text{set-builder notation}} = \{\underbrace{2x}_{\text{The set of all numbers of the form} \ 2x \ } \underbrace{:}_{\text{such that} \ } \underbrace{ \ x \in \mathbb{Z}}_{x \ \text{belongs to the set of integers}} \}

Note that:

\{..., -4, -2, 0, 2, 4, ...\} = \{2x : x \in \mathbb{Z} \}

\mathcal{A}_{\text{set notation}} = \mathcal{A}_{\text{set-builder notation}}

Set-builder Notation: Concept Check

Which of the following represents the elements of this set?

\{x \in \mathbb{Z}: x^2 - 2 = 0\}

\{ \sqrt{2} \}\emptyset\{ 2 \}\{ -\sqrt{2}, \sqrt{2}\}

The \emptyset is the correct answer given that there are no integers x such that x^2 - 2 = 0. The solutions to the equation x^2 - 2 = 0 are x = \sqrt{2} and x = -\sqrt{2}, neither of which are integers. Therefore, the set contains no elements.

Set-builder Notation: LaTeX

The LaTeX command \{2x : x \in \mathbb{Z} \} represents the following set \{2x : x \in \mathbb{Z} \}

Set-builder Notation: Exercises

Set-builder Notation: Exercises 1-5

Write the following elements of the following sets:

\{5x - 1: x \in \mathbb{Z}\}
\{x \in \mathbb{R}: x^2 = 3\}
\{x \in \mathbb{R}: x^2 + 5x = -6\}
\{x \in \mathbb{Z}: \lvert x \rvert < 5\}
\{5a + 2b: a,b \in \mathbb{Z}\}

Set-builder Notation: Exercises 6-10

Write the following sets in set-builder notation:

\{2,4,8,16,32,64\ldots\}
\{\ldots,-6,-3,0,3,6,9,12,15\ldots\}
\{0,1,4,9,16,25,36\ldots\}
\{3,4,5,6,7,8\}
\{\ldots,\frac{1}{8},\frac{1}{4},\frac{1}{2},1,2,4,8,\ldots\}