Subsets are the part of one of the mathematical concepts, Sets. A set is a collection of objects or elements, grouped together in the curly braces, such as {a,b,c,d}. The elements could be anything such as a group of real numbers, variables, constants, whole numbers, etc. It consists of a null set as well.

Subsets are classified as

  • Proper Subset
  • Improper Subsets

A proper subset is one that contains few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.

For example, if set A = {2, 4, 6}, then,

Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.

Proper Subset: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}

Improper Subset: {2,4,6} and Φ

There is no particular formula to find the subsets, instead, we have to list them all, to differentiate between proper and improper one. The set theory symbols was developed by mathematicians to describe the collections of objects.

Definition of Subset

A collection of elements is known as a subset of all the elements of the set are contained inside another set.

Set A is said to be a subset of Set B if all the elements of Set A are also present in Set B.

Example: If set A has {X,Y} and set B has {X,Y,Z}, then A is the subset of B because elements of A are also present in set B.

Subset Symbol

In a set theory, a subset is denoted by the symbol ⊆ and read as ‘is a subset of’.

Using this symbol we can express subsets as follows:

A ⊆ B; which means Set A is a subset of Set B.

Note: A subset can be equal to the set. That is, a subset can contain all the elements that are present in the set.

What is a Proper Subset?

Set A is considered to be a proper subset of Set B, if Set B contains at least one element that is not present in Set A.

Example: If set A has elements as {12, 24} and set B has elements as {12, 24, 36}, then set A is the proper subset of B, because 36 is not present in the set A. To be noticed here, set B is not a subset of set B.

Proper Subset Symbol

A proper subset is denoted by ⊂ and is read as ‘is a proper subset of’. Using this symbol, we can express a proper subset for set A and set B as;

A ⊂ B

How to List Subsets?

If we have to pick n number of elements from a set containing N number of elements, it can be done in NCn number of ways.

Therefore the number of possible subsets containing n number of elements from a set containing N number of elements is equal to NCn.

How many subsets and proper subsets does a set have?

Consider a set having “n” number of elements.

The number of subsets of the set = 2n

Number of Proper Subsets of a set

Since considered set contains ‘n’ elements, then the number of proper subsets of the set is 2n – 1.

Important: Possible subsets of a Set is Set itself but Set is not a proper subset of itself.

If S = {a, b}

Subsets of S = {{}, {a}, {b}, {a,b}}

Number of subsets = 22= 4

Proper subsets of S = {{}, {a}, {b}}

Number of proper subsets = 22 – 1 = 3

What is Improper Subset?

A subset which contains all the elements of the original set is called an improper subset. Basically, the improper set includes the null set and the original set itself. It is denoted by ⊆.

For example: Set P ={2,4,6} Then, the subsets of P are;

{}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} and {2,4,6}.

Where, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} are the proper subsets and {}, {2,4,6} are the improper subsets. Therefore, we can write, {} ⊆ P and {2,4,6} ⊆ P.

Power Set

Power set is said to be the collection of all the subsets. It is represented by P(A).

If A is set having elements{a,b}. Then the power set of A will be;

P(A) =  {∅, {a}, {b}, {a, b}}

To learn more in brief, click on the article link of power set.

Subsets Example Problems

Example 1: How many number of subsets containing three elements can be formed from the set

S = { 1, 2,3,4,5,6,7,8,9,10 }

Solution: Number of elements in the set = 10

Number of elements in the subset = 3

Therefore, the number of possible subsets containing 3 elements = 10C3

Therefore, the number of possible subsets containing 3 elements from the set

S = { 1,2,3,4,5,6,7,8,9,10 } is 120.

Example 2: Given any two real-life examples on the subset.

Solution: We can find a variety of examples of subsets in everyday life such as:

  1. If we consider all the books in a library as one set, then books pertaining to Maths is a subset.
  2. If all the items in a grocery shop form a set, then cereals form a subset.