Systems for Expressing Sets

**Systems for Expressing Sets**

A set is an assortment of distinctive components that do not repeat. Sets can be defined in different forms. Here are four common methods of interpreting them.

**Set-Builder Notation**

Set-builder notation is a mathematical coding system that shows the properties that the components of a set need to satisfy. A simple example is written as

which is expressed as the set ("{ }") of all *a* ("*a*"),such that (" | ") *a* is less than or equal to three ("*a* ≤ 3"), which can also be expressed as any value less than or equal to 3.

**Example**

Describe the following set:

In set-builder notation, this set is written as:

This set is described as all integer numbers that are less than or equal to three and equal to or greater than negative one, or all integer numbers between -1 and 3 inclusive. The bold "Z" in the expression indicates that the *x*-values in the set are limited to the integers.

Here is another example:

This set is described as all integer numbers that are between four and negative two, non-inclusive.

**Roster Notation**

A roster is a list of the elements in a set. They are surrounded by braces and separated by commas.

**Example**

Write in roster notation all the integer numbers that are greater than 0 and less than 10, non-inclusive:

Write in roster notation all integer numbers that are less than two, inclusive.

**Interval and Graphical Notation**

An interval is a connected subset of numbers, an alternative to expressing the values as an inequality. When using interval notation we use two types of symbols:

" **( **" which means non-inclusive or open.

" [ " which means inclusive or closed.

Here’s how you would describe a set with interval notation:

This set represents all numbers between *a* and *b*, including *b* but not including *a*.

You can show an interval on a graph as well. Here are six types of interval notation and how they generally look on a number line:

Notice how interval notation and graphical notation always include all numbers in their sets, not just the integers. For example, 3.2 is an element of the set [2, 5], but not the set {2, 3, 4, 5}.

**Example**

Write this notation for all integers between 23, inclusive, and 28, non-inclusive.

**Roster Notation**

**Set-Builder Notation**

Write this notation for all numbers between 23, inclusive, and 28, non-inclusive.

**Interval Notation**

**Graphical Notation**