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# Describing Sets – Methods & Examples

In mathematics, we deal with different collections of numbers, symbols, or even equations. We give these kinds of collections a special name in mathematics; we call them **sets**. We may wish to **describe** these collections as a way of understanding their properties or discussing their relationships with each other.

You will encounter both large and small sets; therefore, you should learn **how to describe these sets**.

Before we embark on describing sets, it important to learn how to define and write a set.

*In this article, we will learn:*

- How to define, write, and describe a set.
- The key properties of sets.

Remember, we have provided a practice test and an answer key at the end of this article. Don’t forget to test your understanding.

Let’s start by defining a set.

## What is a set in math?

A set is a collection of well-defined objects. We refer to these objects as **members** or **elements** of the set.

Like in ordinary language, we usually talk of sets of cutleries or sets of chairs, etc. In mathematics, we can also talk of sets of numbers, sets of equations, or sets of variables.

For example, the set of natural numbers contains all the natural numbers. Therefore, each natural number is an element or member of that set.

We usually apply the concept of a set as a prerequisite to understanding several branches of mathematics, such as algebra, mathematical analysis, and probability theory.

## How do we write a set in math?

Writing a set in math is pretty simple. We just:

- list the elements in the set,
- separate each element in the set using a comma,
- enclose the elements in the set using curly braces, {}.

For example, the numbers 5,6 and 7 are members of the set {5,6,7}

By convention, we should use an uppercase letter to denote a set and lowercase letters to denote a set’s elements. Also, we should always put an equality sign after the uppercase letter just before writing the elements of the set.

Let’s say we want to write down set A with the elements a, b, and c. So, we will write it as follows:

**A={a,b,c} **

We can also write down set B that has elements 1,2,3, 4, and 5 as follows:

We can also write sets within a set. For example, sets D and E below.

D={p,q,{p,q,r}}

E={1,2,{3,5},6}

Set D contains the set {p,q,r}, and set E contains the set {3,5}.

### Set Membership

We use the symbol ∈ to show that an object is a member of a set. The symbol is read as ‘is an element of’ or ‘is a member of.’

1 is an element of set B above, so we write 1 ∈ B.

We use the symbol ∉ to show that an object is not a member of a set. The symbol is read as ‘is not an element of’ or ‘is not a member of.’

7 is not an element of set B above, so we write 7 ∉ B.

In some cases, we will encounter very large sets or even infinite sets in mathematics. This makes it impossible to list all the elements in the set. In such cases, we:

- write down a few elements of the set to establish the pattern, say, 4 or 5 elements.
- put an ellipsis sign or three dots to show that the set has elements that continue in the same pattern.

We can put the ellipsis sign between the listed elements to show that there are other elements between the listed elements or after the listed elements to show other elements after the ones we have listed. Sets A and N illustrate this.

We write the set A of all the odd numbers between 30 and 70 as:

**A**={31,33,35,…,67,69}

We also write the set N of all the natural numbers as:

**N**={1,2,3,4,…}

## Properties of sets

We consider these properties when writing down sets.

**A set must be well defined**.

This eliminates the chances of ambiguity. For example, ‘the set of all short people’ is not well defined, but ‘the set of all people with a height less than 5.5 feet’ is well defined.

**The elements of a given set must be distinct.**

Elements in a set should not be repeated. For example, we should write the set {1,3,5,3,7,9,7} as {1,3,5,7,9}.

The order in which the elements are written in a set does not matter. For example, the set {1,2,3,4} can be written as {4,3,2,1}, or {2,4,3,1}. All these sets are the same.

Now, we can comfortably learn how to describe sets.

## How do we describe a set?

When we specify elements of a set, we are simply describing the set. The most common methods used to describe sets are:

- The verbal description method
- The roster notation or listing method
- The set-builder notation

Let us go into the details.

### The verbal description method

When using this method, we describe the set in words using a verbal statement. We have to ensure that the statement is well-defined.

Examples of sets written using the verbal description method:

- The set of colors on the American flag.
- The set of all the natural numbers less than 10.
- The set of all even numbers.
- The set of all integers between -10 and -15.

### The roster notation or listing method

This method is also called the tabulation method. When using this method, we list the elements of the set in a row between curly braces.

We refer to this method as the roster notation because a roster is a list of elements in the set.

This method is also known as the **enumeration method** because we usually list the elements, one after the other.

We should always separate the elements using commas.

This method is convenient when describing small sets.

#### Limitations of the roster notation

The roster notation is a straightforward method of describing sets but not convenient when describing large sets. Imagine using the roster method to describe the set of all the natural numbers less than 100!

**Examples of sets written using the roster notation:**

Now, let’s convert the sets above from the verbal description method to roster notation.

A={white,red,blue}

B={1,2,3,4,5,6,7,8,9}

C={2,4,6,8,….}

D={-11,-12,-13,-14}

## The set-builder notation

When using this method, we:

- set a variable to represent any element in the set.
- add a brief description of a specific property that is common to all members of that set.

We have to ensure that the property we are using to describe the elements of the set should be common to all the elements in that set. This helps us to tell clearly which objects belong to the set and which ones do not.

We can describe set K, using the set-builder notation as shown below.

K={*x*|* x* has the property M} or

K={*x*: *x* has the property M}, where *x* is the set variable

We read this as **‘set K is the set of all elements x, such that x has the property M.’**

The vertical bar (|) or the colon (:) can be used interchangeably to replace the phrase **‘such that’** or **‘for which’** when describing sets. We use either the vertical bar or the colon to separate the variable we have set from the property we are using to describe the elements of the set.

#### The advantage of the set-builder notation

The set-builder notation is more suitable than the roster notation because it can be used to describe both large and small sets.

Let’s use the set-builder notation to describe the set T of all integers greater than 5.

We select *y* as our set variable and identify a suitable property that describes the set. In this case, *y* must be an integer greater than 5.

We describe set T as shown below:

T={*y*| *y* is an integer,*y>5*}

Let’s convert the examples above into the set-builder notation.

**Examples of sets written using the set-builder notation**

A={*x| x* is a color of the American flag }

B={*y*:*y* is a natural number less than 10}

C={*x*:*x* is an even number}

D={*m*|*m* is an integer between -10 and -15 }

We can also use the set-builder notation to describe intervals of real numbers, as shown in the table below.

Interval | Description |

[a,b] | {x| a≤x≤b} (closed interval) |

(a,b] | {x| a<x≤b} (half-open interval) |

[a,b) | {x| a≤x<b} (half-open interval) |

(a,b) | {x| a<x<b} (open interval) |

#### Different Methods of Describing Sets

Verbal description | Set-builder notation | Roster notation |

The set of all odd positive numbers less than or equal to 5 | {x:x is an odd number and 0<x≤5} | {1,2,3,4,5} |

#### Descriptions of Sets of Numbers in Mathematics

The table below shows some of the sets of numbers you may encounter in the course of studying mathematics.

Set name | Symbol | Description |

Natural numbers | N | N={1,2,3,…} N={x| x is a natural number} |

Whole numbers | W | W={0,1,2,3,…} W={x| x is a whole number} |

Integers | Z | Z={…,-3,-2,-1,0,1,2,3,…} Z={x| x is an integer} |

Rational numbers | Q | Q={x| x is a rational number} Q={x| x can be written in the form p/q where q≠0} |

Real numbers | R | R={x| x is a real number} |

Complex numbers | C | C={x: x is a complex number} C={x+yi| a,b∈R and i is an imaginary unit } |

Thus far, we have had so much fun describing sets. Now, it’s time to try out a few questions.

*Practice Questions*

*Practice Questions*

- Describe set A containing all natural numbers less than 10 using:

(a) The set-builder notation

(b) The roster notation - Describe the set M below using the verbal description method.
**M**={*x*|*x*∈R,0<*x*<1} - Describe the set N using the set-builder notation.

N={1,3,5,7,9} - Write down the set E of positive even numbers less than 10 using the roster notation.
- Describe the set P of all prime numbers greater than 100 using the set-builder notation.

#### Answer key

- (a) A={
*x*|*x*is a natural number less than 10}/ A={x| x∈N,x<10}/A={*x*|*x*is a natural number and x<10} (b) A={1,2,3,4,5,6,7,8,9} - The set M is the set of all real numbers between 0 and 1.
- N={
*x*|*x*is a positive odd number less than 10}/N={*x*|*x*is a positive odd number and x<10} - E={2,4,6,8}
- P={
*x*|*x*is a prime number greater than 100}/P={*x*|*x*is a prime number and x>100}