**Set-builder & Interval Notation**

A **set** is a collection of unique elements. Elements in a set do not “repeat”.

**Methods of Describing Sets:**

Sets may be described in many ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and/or by Venn diagrams. For graphing on a number line, see Linear Inequalities. For Venn diagrams, see Working with Sets and Venn Diagrams.

**By roster:** A roster is a list of the elements in a set, separated by commas and surrounded by French curly braces.

**By set-builder notation:** Set-builder notation is a mathematical shorthand for precisely stating all numbers of a specific set that possess a specific property.

**By interval notation:** An **interval** is a connected subset of numbers. **Interval notation** is an alternative to expressing your answer as an inequality. Unless specified otherwise, we will be working with real numbers.

**Intervals**

**There are four types of interval:**

**Open interval:**Let a and b be two real numbers such that a<b, then the set of all real numbers lying strictly between a and b is called an open interval and is denoted by ]a, b[ or (a, b). Thus, ]a, b[ or (a, b)= { x ∈ R: a < x < b}.**Closed interval:**Let a and b be two real numbers such that a<b, then the set of all real numbers lying between a and b including a and b is called a closed interval and is denoted by [a, b]. Thus, [a, b] = { x ∈ R: a ≤ x ≤ b}.**Open-Closed interval:**It is denoted by ]a, b] or (a, b] and ]a, b] or (a, b] = { x ∈ R: a < x ≤ b}.**Closed-Open interval:**It is denoted by [a, b[ or [a, b) and [a, b[ or [a, b) = { x ∈ R: a ≤ x < b}.

The chart below will show you all of the possible ways of utilizing interval notation.

For some intervals it is necessary to use combinations of interval notations to achieve the desired set of numbers. Consider how you would express the interval **“all numbers except 13”.**

Consider expressing in interval notation, the set of numbers which contains all numbers less than 0 and also all numbers greater than 2 but less than or equal to 10.

As you have seen, there are many ways of representing the same interval of values. These ways may include word descriptions or mathematical symbols.