Types of Sets (advance math)

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Types of Sets


What are the different types of sets?

The different types of sets are explained below with examples.


Empty Set or Null Set:

A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.

For example: (a) The set of whole numbers less than 0.

(b) Clearly there is no whole number less than 0.

Therefore, it is an empty set.

(c) N = {x : x ∈ N, 3 < x < 4}

Let A = {x : 2 < x < 3, x is a natural number}

Here A is an empty set because there is no natural number between
2 and 3.

Let B = {x : x is a composite number less than 4}.

Here B is an empty set because there is no composite number less than 4.


∅ ≠ {0} ∴ has no element.

{0} is a set which has one element 0.

The cardinal number of an empty set, i.e., n(∅) = 0


Singleton Set:

A set which contains only one element is called a singleton set.

For example:

A = {x : x is neither prime nor composite}

It is a singleton set containing one element, i.e., 1.

B = {x : x is a whole number, x < 1}

This set contains only one element 0 and is a singleton set.

Let A = {x : x ∈ N and x² = 4}

Here A is a singleton set because there is only one element 2 whose square is 4.

Let B = {x : x is a even prime number}

Here B is a singleton set because there is only one prime number which is even, i.e., 2.


Finite Set:

A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.

For example:

The set of all colors in the rainbow.

N = {x : x ∈ N, x < 7}

P = {2, 3, 5, 7, 11, 13, 17, …… 97}

Infinite Set:

The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.

For example:

 Set of all points in a plane

 A = {x : x ∈ N, x > 1}

 Set of all prime numbers

 B = {x : x ∈ W, x = 2n}


All infinite sets cannot be expressed in roster form.

For example:

The set of real numbers since the elements of this set do not follow any particular pattern.

Cardinal Number of a Set:

The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by n(A).

For example:

 A {x : x ∈ N, x < 5}

A = {1, 2, 3, 4}

Therefore, n(A) = 4

 B = set of letters in the word ALGEBRA

B = {A, L, G, E, B, R}

Therefore, n(B) = 6

Equivalent Sets:

Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

For example:

A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3

Therefore, A ↔ B

Equal sets:

Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.

For example:

A = {p, q, r, s}

B = {p, s, r, q}

Therefore, A = B

The various types of sets and their definitions are explained above with the help of examples.


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