Types of Sets in Mathematics

Sets are the collection of different elements belonging to the same category and there can be different types of sets seen. A set may have an infinite number of elements, may have no elements at all, may have some elements, may have just one element, and so on. Based on all these different ways, sets are classified into different types.

The different types of sets are:

Let’s discuss these various types of sets in detail.

Singleton Set

Singleton Sets are those sets that have only 1 element present in them.

Example: 

• Set A= {1} is a singleton set as it has only one element, that is, 1.
• Set P = {a : a is an even prime number} is a singleton set as it has only one element 2.

Similarly, all the sets that contain only one element are known as Singleton sets.

Empty Set

Empty sets are also known as Null sets or Void sets. They are the sets with no element/elements in them. They are denoted as ϕ.

Example:

• Set A= {a: a is a number greater than 5 and less than 3}
• Set B= {p: p are the students studying in class 7 and class 8}

Finite Set

Finite Sets are those which have a finite number of elements present, no matter how much they’re increasing number, as long as they are finite in nature, They will be called a Finite set.

Example: 

• Set A= {a: a is the whole number less than 20}
• Set B = {a, b, c, d, e}

Infinite Set

Infinite Sets are those that have an infinite number of elements present, cases in which the number of elements is hard to determine are known as infinite sets. 

Example: 

• Set A= {a: a is an odd number}
• Set B = {2,4,6,8,10,12,14,…..}

Equal Set

Two sets having the same elements and an equal number of elements are called equal sets. The elements in the set may be rearranged, or they may be repeated, but they will still be equal sets.

Example:

• Set A = {1, 2, 6, 5}
• Set B = {2, 1, 5, 6}

In the above example, the elements are 1, 2, 5, 6. Therefore, A= B.

Equivalent Set

Equivalent Sets are those which have the same number of elements present in them. It is important to note that the elements may be different in both sets but the number of elements present is equal. For Instance, if a set has 6 elements in it, and the other set also has 6 elements present, they are equivalent sets.

Example:

Set A= {2, 3, 5, 7, 11}

Set B = {p, q, r, s, t}

Set A and Set B both have 5 elements hence, both are equivalent sets.

Subset

Set A will be called the Subset of Set B if all the elements present in Set A already belong to Set B. The symbol used for the subset is ⊆

If A is a Subset of B, It will be written as A ⊆ B

Example:

Set A= {33, 66, 99}

Set B = {22, 11, 33, 99, 66}

Then, Set A ⊆ Set B 

Power Set

Power set of any set A is defined as the set containing all the subsets of set A. It is denoted by the symbol P(A) and read as Power set of A.

For any set A containing n elements, the total number of subsets formed is 2ⁿ. Thus, the power set of A, P(A) has 2ⁿ elements.

Example: For any set A = {a,b,c}, the power set of A is?

Solution:

Power Set P(A) is,

P(A) = {ϕ, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Universal Set 

A universal set is a set that contains all the elements of the rest of the sets. It can be said that all the sets are the subsets of Universal sets. The universal set is denoted as U.

Example: For Set A = {a, b, c, d} and Set B = {1,2} find the universal set containing both sets.

Solution:

Universal Set U is,

U = {a, b, c, d, e, 1, 2}

Disjoint Sets

For any two sets A and B which do have no common elements are called Disjoint Sets. The intersection of the Disjoint set is ϕ, now for set A and set B A∩B =  ϕ. 

Example: Check whether Set A ={a, b, c, d} and Set B= {1,2} are disjoint or not.

Solution:

Set A ={a, b, c, d}
Set B= {1,2}

Here, A∩B =  ϕ

Thus, Set A and Set B are disjoint sets.

Also, Check

• Set Theory
• Set Theory Symbols
• Relations and Functions
• Representation of a Set
• Operations on Sets

Sets are a well-defined collection of objects. Objects that a set contains are called the elements of the set. We can also consider sets as collections of elements that have a common feature. For example, the collection of even numbers is called the set of even numbers.