Set theory is a fundamental concept throughout all of mathematics. This branch of mathematics forms a foundation for other topics. A set is a collection of objects, which are called elements. Various types of sets based on the type of elements present are studied in Mathematics.

Let’s understand different types of sets with their properties and examples.

## Types of Sets

Types of sets in math are important to learn not only to understand the theories in math but to also apply them in day-to-day life as arranging objects that belong to the alike category and keeping them in one group helps to find things easily and look clean as well.

The following are the different types of sets based on the elements present in them.

### 1. Empty Set

The set, which has no elements, is also called an empty set. It is also called a null set or a void set. It is denoted by two curly braces without any element ‘$\{ \}$’ It is also denoted by the symbol ‘$\phi$’.

#### Examples of Empty Set

**Example 1:** A set $\text{X } = \{x : 1 \lt x \lt 2, x \text{ is a natural number} \}$, is a null or an empty set, as we know a natural number cannot be a decimal or a fractional number.

**Example 2:** A set $\text{S } = \{x : x^2 – 2 = 0, x \text{ is a natural number} \}$, is a null or an empty set, as we know that solutions of $x^2 – 2 = 0$ are irrational numbers and not natural numbers.

**Example 3:** A set $\text{C } = \{s : s \text{ is a student studying in class 6th and class 7th} \}$, is a null or an empty set, as someone cannot be a student of two classes at the same time.

### 2. Singleton Set

A set that has just one element is named a singleton set.

#### Examples of Singleton Set

**Example 1:** A set $\text{P } = \{x : x \text{ is an even prime number} \}$ is a singleton set, as we know that $2$ is the only prime number that is also an even number.

**Example 2:** A set $\text{E} = \{x : x \in \text{N} \text{ and } x^3 = 27 \}$ is a singleton set with a single element $3$.

**Example 3:** A set $\text{W} = \{v: v \text{ is a vowel letter and } v \text{ is the first alphabet of English} \}$ is also a singleton set with just one element $a$.

### 3. Finite and Infinite Sets

A set that has a finite number of elements is known as a finite set, whereas a set whose elements can’t be estimated, but has some figure or number, which is large to precise in a set, is known as an infinite set.

#### Examples of Finite and Infinite Sets

**Example 1:** A set $\text{A} = \{x: x \text{ is a natural number less than or equal to } 10 \}$ is a finite set as the number of elements in the set is $10$. The cardinal number of the set is $n(\text{A}) = 10$.

**Example 2:** A set $\text{F} = \{x: x \text{ is a factor of } 12 \}$ is a finite set. We know that factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$, which are $6$ in number and hence can be counted. The cardinal number of the set is $n(\text{F}) = 6$.

**Example 3:** A set $\text{C} = \{ \text{ number of cattle in India} \}$ is an infinite set, there is an approximate number of cattle in India, but the actual number of cattle cannot be expressed, as the numbers could be very large and counting all cattle is not possible.

**Example 4:** A set $\text{N} = \{x: x \text{ is a natural number}\}$ is also an infinite set, as there are countless natural numbers and hence cannot be counted.

### 4. Equivalent Sets

If the number of elements of set $\text{A}$ is equal to the number of elements of set $\text{B}$, then sets $\text{A}$ and $\text{B}$ are called equivalent sets. In other words, if the cardinal number of sets $\text{A}$ and $\text{B}$ are the same, the sets $\text{A}$ and $\text{B}$ are called equivalent sets.

The two sets $\text{A}$ and $\text{B}$ are equivalent sets, if $n(\text{A}) = n(\text{B})$.

#### Examples of Equivalent Sets

**Example 1:** The sets $\text{A} = \{a, e, i, o, u \}$ and $\text{N} = \{1, 2, 3, 4, 5 \}$ are equivalent sets, as we see that $n(\text{A}) = n(\text{B}) = 5$.

**Example 2:** The sets $\text{A} = \{x: x \text{ is a factor of } 5 \}$ and $\text{B} = \{y: y \text{ is a factor of } 7 \}$ are equivalent sets, as $n(\text{A}) = n(\text{B}) = 2$. (Both $5$ and $7$ are prime numbers and have $2$ factors each).

### 5. Equal Sets

If every element of set $\text{A}$ is also the element of set $\text{B}$ and if every element of set $\text{B}$ is also the element of set $\text{A}$, then sets $\text{A}$ and $\text{B}$ are called equal sets. It means set $\text{A}$ and set $\text{B}$ have equivalent elements and that we can denote it as $\text{A} = \text{A}$.

#### Examples of Equal Sets

**Example 1:** If set $\text{A} = \{3,4,5,6 \}$ and $\text{B} = \{6,5,4,3 \}$, then $\text{A} = \text{B}$.

**Note:** The order of elements in a set does not matter.

**Example 2:** If $\text{A} = \{x | x \text{ is an even number} \}$ and $\text{A} = \{x | x \text{ is a natural number} \}$ then $\text{A} \ne \text{B}$, because natural numbers consist of all the positive integers starting from $1$, $2$, $3$, $4$, $5$ to infinity, but even numbers start with $2$, $4$, $6$, $8$, and so on.

**Note:** All equal sets are equivalent sets, but all equivalent sets are not equal sets.

### 6. Subsets

A set $\text{S}$ is said to be a subset of set $\text{T}$ if the elements of set $\text{S}$ belong to set $\text{T}$, or you can say each element of set $\text{S}$ is present in set $\text{T}$. A subset of a set is denoted by the symbol $\subset$ and written as $\text{S} \subset \text{T}$.

We can also write the subset notation as $\text{S} \subset \text{T}$ if $p \in \text{S} => p \in \text{T}$.

According to the above statements, “$\text{S}$ is a subset of $\text{T}$ only if $p$ is an element of $\text{S}$ as well as an element of $\text{T}$.” Each set is a subset of its own set, and a void set or empty set is a subset of all sets.

#### Examples of Subsets

**Example 1:** If $\text{N} = \{x: x \text{ is a natural number} \}$, $\text{E} = \{y: y \text{ is an odd number} \}$, and $\text{O} = \{x: x \text{ is an odd number} \}$, then $\text{E} \subset \text{N}$, and also $\text{O} \subset \text{N}$.

### 7. Superset

A set $\text{T}$ is said to be a superset of set $\text{S}$ if the elements of set $\text{S}$ belong to set $\text{T}$, or you can say each element of set $\text{S}$ is present in set $\text{T}$. A superset of a set is denoted by the symbol $\supset$ and written as $\text{T} \supset \text{S}$.

We can also write the superset notation as $\text{T} \supset \text{T}$ if $p \in \text{S} => p \in \text{T}$.

According to the above statements, “$\text{T}$ is a subset of $\text{S}$ only if $p$ is an element of $\text{S}$ as well as an element of $\text{T}$.” Each set is a subset of its own set, and a void set or empty set is a subset of all sets.

#### Examples of Subsets

**Example 1:** If $\text{N} = \{x: x \text{ is a natural number} \}$, $\text{E} = \{y: y \text{ is an odd number} \}$, and $\text{O} = \{x: x \text{ is an odd number} \}$, then $\text{N} \supset \text{E}$, and also $\text{N} \supset \text{O}$.

### 8. Power Set

The set of all subsets is known as a power set. We know the empty set is a subset of all sets, and each set is a subset of itself. Taking an example of set $\text{X} = \{2,3 \}$. From the above-given statements, we can write, $\{ \}$ is a subset of $\text{2,3 \}$.

#### Examples of Power Set

**Example 1:** $\{2 \}$ is a subset of $\{2,3 \}$.

**Example 2:** If $\text{N} = \{x: x \text{ is a natural number} \}$, $\text{E} = \{y: y \text{ is an odd number} \}$, and $\text{O} = \{x: x \text{ is an odd number} \}$, then $\text{N}$ is a power set of both the sets $\text{E}$, and $\text{O}$.

**Note:** A set of Real Numbers is a power set of a set of natural numbers, a set of integers, a set of rational numbers, and a set of irrational numbers.

### 9. Universal Set

A set that contains all the elements of other sets is called a universal set. Generally, it is represented as $\text{U}$.

#### Examples of Universal Set

**Example 1:** If set $\text{A} = \{1,2,3 \}$, set $\text{B} = \{3,4,5,6 \}$, and $\text{C} = \{5,6,7,8,9 \}$. Then, we will write the universal set as, $\text{U} = \{1,2,3,4,5,6,7,8,9 \}$.

**Note:** According to the definition of the universal set, we can say that all the sets are subsets of the universal set. Therefore, $\text{A} \subset \text{U}$, $\text{B} \subset \text{U}$, and $\text{C} \subset \text{U}$.

### 10. Disjoint Sets

If two sets $\text{X}$ and $\text{Y}$ do not have any common elements, and their intersection results in an empty set $\phi$, then set $\text{X}$ and $\text{Y}$ are called disjoint sets. It can be represented as $\text{X} \cap \text{Y} = \phi$.

#### Examples of Disjoint Sets

**Example 1:** If $\text{E} = \{x: x \text{ is an even number} \}$, and $\text{O} = \{x: x \text{ is an odd number} \}$, then sets $\text{E}$ and $\text{O}$ are disjoint sets, i.e., $\text{E} \cap \text{O} = \phi$.

## Set Symbols

Set symbols are used to define the elements of a given set. The following presents some of these symbols with their meaning.

Symbol | Meaning |

$\cup$ | Universal set |

$n(\text{A})$ | Cardinal number of set $\text{A}$ |

$c \in \text{B}$ | $c$ belongs to $\text{B}$, or $c$ is an element of set $\text{B}$ |

$a \notin \text{A}$ | $a$ does not belongs to $\text{A}$, or $a$ is not an element of set $\text{A}$ |

$\phi$ | An empty, null, or void set |

$\text{A} \subset \text{B}$ | Set $\text{A}$ is a subset of set $\text{B}$ |

$\text{X} \supset \text{Y}$ | Set $\text{X}$ is a superset of set $\text{Y}$ |

## Practice Problem

- If $\text{A} = \{1,2,3,4,5,7,9,11 \}$, find $n(\text{A})$.
- Write an example of a finite and infinite set in set builder form.
- Write an example of equal sets.
- Are $\text{P} = \{ x : –3 \le x \le 0, x \in \text{Z} \}$ and $\text{Q} $= The set of all prime factors of $210$, equivalent sets?
- Are $\text{A} = \{x : x \in \text{N}, 4 \le x \le 8 \}$ and $\text{B} = \{ 4, 5, 6, 7, 8 \}$ equal sets?
- Write all the subsets of $\text{A} = \{a, b, c \}$.
- Insert the appropriate symbol $\subset$ or $\not\subset$ in each blank to make a true statement.
- $\{10, 20, 30 \}$ ____ $\{10, 20, 30, 40 \}$
- $\{p, q, r \}$ _____ $\{w, x, y, z \}$

- Find the number of subsets and the number of proper subsets of a set $\text{X}=\{a, b, c, x, y, z \}$.
- If set $\text{A} = \{1, 3, 5 \}$, $\text{B} = \{2, 4, 6 \}$ and $\text{C} = \{0, 2, 4, 6, 8 \}$. Then write the universal set for all three sets.

## FAQs

### What are the different types of sets?

The different types of sets are Empty Sets, Singleton Set, Finite and Infinite Sets, Equivalent Sets, Equal Sets, Subsets, Superset, Power Set, Universal Set, and Disjoint Sets.

### What is called a set?

A set is a collection of objects, which are called elements. Various types of sets based on the type of elements present are studied in Mathematics.

### What are finite and infinite sets?

Any set that is empty or consists of a definite and countable number of elements is referred to as a finite set. Whereas, sets with uncountable or indefinite numbers of elements are called infinite sets.

### What is a universal set?

A universal set consists of all the elements of a problem under consideration. We generally represent it by the letter U. For instance, the set of real numbers is a universal set of all-natural, whole, odd, even, and rational in addition to irrational numbers.

### What are equal and unequal sets?

Two sets $\text{A}$ and $\text{B}$ are said to be equal if they have exactly similar elements. Here the elements can be irrespective of the order of appearance in the set. Equal sets are represented as $\text{A} = \text{B}$. Otherwise, the sets are referred to as unequal sets, which are represented as $\text{A} \ne \text{B}$.

For example, if $\text{A} = \{1, 2, 3 \}$ and $\text{D} = \{1, 3, 2 \}$ then both of these sets are equal.

### Is an Empty set a type of Finite set?

Yes, an empty set is a type of finite set as it contains no elements. The number of elements in an empty set is definite, that is, zero, therefore, it is a finite set. The cardinal number of an empty set is $0$, i.e., if $\text{A}$ is an empty set, then $n\text{A} = 0$.

## Conclusion

A set is a collection of objects, which are called elements. Various types of sets based on the type of elements present are studied in Mathematics. The different types of sets are Empty Sets, Singleton Set, Finite and Infinite Sets, Equivalent Sets, Equal Sets, Subsets, Superset, Power Set, Universal Set, and Disjoint Sets.