Introduction

The domain of a set is a description of the elements it contains. There are multiple methods in which to notate and describe a set, in this article we will explore: roster method, set-builder notation, and interval notation.

Roster Method

In roster method we notate a set by setting it equal to a list of its member elements contained within a pair of brackets. For example, to describe a set S as all integers between 3 and 7, we write:

While this method is simple and straightforward, the downside is the burden it becomes to write large sets. In some cases (as those that involve infinity), we are forced to describe the set using another method like Set-Builder Notation.

Set-Builder Notation

In set-builder notation, we describe the domain of a set between two brackets. To describe a set with a finite list of elements we separate the elements with either commas or semicolons:

To describe a range of values (or all numbers that exist between two endpoints) we use inequalities. First we declare the variable which represents the range. Next, we draw a vertical line to separate the declaration from the definition. Finally, we place inequality before adding the final bracket. To describe a set containing all values greater than seven we write:

In the previous example we excluded the endpoint 7, but if we want to include it we say, "all values greater than or equal to seven."

To describe a value between two values we can merge the two inequalities. To describe a range that is "greater than three and less than or equal to seven" we write:

We can use commas and set operations to describe a set with multiple ranges (or a range of multiple parts). For example, given the two ranges:

• "greater than or equal to zero and less than three"
• "greater than four and less than seven"

We can either separate the two ranges with a comma (or semicolon):

Or join them using the union operation (can also use the 'and' operator):

As we can see, operating on multiple ranges using set-builder notation can become cumbersome. Set interval notation is used to simplify the description and operation on a range.

Interval Notation

Set interval notation is used to condense the notation of a range of values. Instead of the variable-inequality pairs used in set-builder notation, we can simplify the description down to just the two endpoints and whether or not either is included. In set interval notation, we list the two endpoints separated with a comma; the left being the lesser.

To describe the range of values that are "greater than three and less than or equal to seven" with interval notation, we write:

This is equivalent to its pair in set-builder notation {x|3<x≤7}. We place a parenthesis by endpoints which are excluded in the range (as with the three). Likewise, we place a bracket next to endpoints which are included (like the seven).

Endpoints that stretch to infinity (∞) are always excluded (or marked with parentheses) because they are never reached. For example, to represent the range "all values greater than seven {x|x>7} we write:

Learn how to illustrate the domain using the Real Number Line.