# Set Notation – Explanation & Examples

Set notation is used to define the elements and properties of sets using symbols. Symbols save you space when writing and describing sets.

Set notation also helps us to describe different relationships between two or more sets using symbols. This way, we can easily perform operations on sets, such as unions and intersections.

You can never tell when set notation will show up, and it can be in your algebra class! Therefore, knowledge of the symbols used in set theory is an asset.

• How to define a set notation
• How to read and write set notation

You will find a short quiz accompanied by an answer key at the end of this article. Don’t forget to test how much you’ve grasped.

## What is set notation?

Set notation is a system of symbols used to:

• define elements of a set
• illustrate relationships among sets
• illustrate operations among sets

In the previous article, we used a few of these symbols when describing sets. Do you remember the symbols shown in the table below?

 Symbol Meaning ∈ ‘is a member of’ or ‘is an element of’ ∉ ‘is not a member of’ or ‘is not an element of’ { } denotes a set | ‘such that’ or ‘for which’ : ‘such that’ or ‘for which’

Let’s introduce more symbols and learn how to read and write these symbols.

## How do we read and write set notation?

To read and write set notation, we need to understand how to use symbols in the following cases:

#### 1. Denoting a Set

Conventionally, we denote a set by a capital letter and denote the elements of the set by lower-case letters.

We usually separate the elements using commas. For example, we can write the set A that contains the vowels of the English alphabet as:

We read this as ‘the set A containing the vowels of the English alphabet’.

#### 2. Set Membership

We use the symbol ∈ is used to denote membership in a set.

Since 1 is an element of set B, we write 1∈B and read it as ‘1 is an element of set B’ or ‘1 is a member of set B’.
Since 6 is not an element of set B, we write 6∉B and read it as ‘6 is not an element of set B’ or ‘6 is not a member of set B’.

#### 3. Specifying Members of a Set

In the previous article on describing sets, we applied set notation in describing sets. I hope you still remember the set-builder notation!

We can describe set B above using the set-builder notation as shown below:

We read this notation as ‘the set of all x such that x is a natural number less than or equal to 5’.

#### 4. Subsets of a set

We say that set A is a subset of set B when every element of A is also an element of B. We can also say that A is contained in B. The notation for a subset is shown below:

The symbol stands for ‘is a subset of’ or ‘is contained in.’ We usually read A⊆B as ‘A is a subset of B’ or ‘A is contained in B.’
We use the notation below to show that A is not a subset of B:

The symbol stands for ‘is not a subset of’; therefore, we read A⊈B as ‘A is not a subset of B.’

#### 5. Proper Subsets of a Set

We say that set A is a proper subset of set B when every element of A is also an element of B, but there is at least one element of B that is not in A.

We use the notation below to show that A is a proper subset of B:

The symbol stands for ‘proper subset of’; therefore, we read A⊂B as ‘A is a proper subset of B.’

We refer to B as the superset of  A. The figure below illustrates A as a proper subset of  B and B as the superset of A.

#### 6. Equal Sets

If every element of set A is also an element of set B, and every element of B is also an element of A, then we say that set A is equal to set B.

We use the notation below to show that two sets are equal.

We read A=B as ‘set A is equal to set B’ or ‘set A is identical to set B.’

#### 7. The Empty Set

The empty set is a set that has no elements. We can also call it a null set. We denote the empty set by the symbol ∅ or by empty curly braces, {}.

It is also worth noting that the empty set is a subset of every set.

#### 8. Singleton

A singleton is a set that contains exactly one element. Due to this reason, we also call it a unit set. For example, the set {1} contains only one element,1.

We enclose the single element in curly braces to denote a singleton.

#### 9. The Universal Set

The universal set is a set that contains all the elements under consideration. Conventionally, we use the symbol U to denote the universal set.

#### 10. The Power Set

The power set of set A is the set that contains all the subsets of A. We denote a power set by P(A) and read it as ‘the power set of A.’

#### 11. The Union of Sets

The union of set A and set B is the set that contains all elements in set A or set B or in both set A and set B.

We denote A and B’s union by A ⋃ B and read it as ‘A union B.’ We can also use the set-builder notation to define the union of A and B, as shown below.

The union of three or more sets contains all the elements in each of the sets.
An element belongs to the union if it belongs to at least one of the sets.
We denote the union of the sets B1, B2, B3,…., Bn by:

The figure below shows the union of set A and set B.

Example 1
If A={1,2,3,4,5} and B={1,3,5,7,9} then A∪B={1,2,3,4,5,7,9}

#### 12. The Intersection of Sets

The intersection of set A and set B is the set containing all the elements that belong to both A and B.

We denote A and B’s intersection by A ∩ B and read it as ‘A intersection B.’
We can also use the set-builder notation to define A and B’s intersection, as shown below.

The intersection of three or more sets contains elements that belong to all the sets.
An element belongs to the intersection if it belongs to all the sets.
We denote the intersection of the sets B1, B2, B3,…., Bn by:

The figure below shows the intersection of set A and set B illustrated by the shaded region.

Example 2
If A={1,2,3,4,5} and B={1,3,5,7,9} then A∩B={1,3,5}

#### 13. The Complement of a Set

14The complement of set A is a set that contains all elements in the universal set that are not in A.

We denote the complement of set A by Ac or A’. The complement of a set is also called the absolute complement of the set.

#### 14. Set Difference

The set difference of set A and set B is the set of all elements found in A but not in B.

We denote A and B’s set difference by A\B or A-B and read it as ‘A difference B.’

The set difference of A and B is also called the relative complement of B with respect to A.

Example 3
If A={1,2,3} and B={2,3,4,5} then A\B=A-B={1}

#### 15. The cardinality of a Set

The cardinality of a finite set A is the number of elements in A.
We denote the cardinality of set A by |A| or n(A).

Example 4
If A={1,2,3}, then |A|=n(A)=3 because it has three elements.

#### 16. The Cartesian Product of Sets

The Cartesian product of two non-empty sets, A and B, is the set of all ordered pairs (a,b) such that a∈A and b∈B.

We denote the Cartesian product of A and B by A×B.

We can use the set-builder notation to denote the Cartesian product of A and B, as shown below.

Example 5
If A={5,6,7} and B={8,9} then A×B={(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)}

#### 17. Disjoint Sets

We say that sets A and B are disjoint when they do not have any element in common.

The intersection of disjoint sets is the empty set.
If A and B are disjoint sets, then we write:

Example 6
If A={1,5}, and B={7,9} then A and B are disjoint sets.

## Symbols Used in Set Notation

Let’s summarize the symbols we have learned in the table below.

 Notation Name Meaning A∪B Union Elements that belong to set A or set B or both A and B A∩B Intersection Elements that belong to both set A and set B A⊆B Subset Every element of set A is also in set B A⊂B Proper subset Every element of A is also in B, but B contains more elements A⊄B Not a subset Elements of set A are not elements of set B A=B Equal sets Both set A and B have the same elements Ac or A’ Complement Elements not in set A but in the universal set A-B or A\B Set difference Elements in set A but not in set B P(A) Power set The set of all subsets of set A A×B Cartesian product The set that contains all the ordered pairs from set A and B in that order n(A) or |A| Cardinality The number of elements in set A ∅ or { } Empty set The set that has no elements U Universal set The set that contains all the elements under consideration N The set of natural numbers N={1,2,3,4,…} Z The set of integers Z={…,-2,-1,0,1,2,…} R The set of real numbers R={x|-∞
 C The set of complex numbers C={z|z=a+bi and a,b∈R and i=√(-1)}

### Practice Questions

Consider the three sets below:
U={0,4,7,9,10,11,15}
A={4,7,9,11}
B={0,4,10}
Find:

1. A∪B
2. A∩B
3. n(A)
4. P(A)
5. |B|
6. A-B
7. Bc
8. A×B