Subsets

Subsets are a part of one of the mathematical concepts called Sets. A set is a collection of objects or elements, grouped in the curly braces, such as {a,b,c,d}. If a set A is a collection of even number and set B consists of {2,4,6}, then B is said to be a subset of A, denoted by B⊆A and A is the superset of B. Learn Sets Subset And Superset to understand the difference.

The elements of sets could be anything such as a group of real numbers, variables, constants, whole numbers, etc. It consists of a null set as well. Let us discuss subsets here with its types and examples.

Table of contents:

• Definition
  • Symbol
  • All subsets
• Types
• Proper Subset
  • • Proper Subset Symbol
    • Formula
  • Subsets and Proper Subsets
• Improper Subsets
  • Power set
• Properties
• Solved Examples
• FAQs

What is a Subset in Maths?

Set A is said to be a subset of Set B if all the elements of Set A are also present in Set B. In other words, set A is contained inside 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 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.

All Subsets of a Set

The subsets of any set consists of all possible sets including its elements and the null set. Let us understand with the help of an example.

Example: Find all the subsets of set A = {1,2,3,4}

Solution: Given, A = {1,2,3,4}

Subsets =

{}

{1}, {2}, {3}, {4},

{1,2}, {1,3}, {1,4}, {2,3},{2,4}, {3,4},

{1,2,3}, {2,3,4}, {1,3,4}, {1,2,4}

{1,2,3,4}

Types of Subsets

Subsets are classified as

• Proper Subset
• Improper Subsets

A proper subset is one that contains a 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 Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}

Improper Subset: {2,4,6}

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 were developed by mathematicians to describe the collections of objects.

What are Proper Subsets?

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.

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

Proper Subset Formula

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

How many subsets and proper subsets does a set have?

If a set has “n” elements, then the number of subset of the given set is 2ⁿ and the number of proper subsets of the given subset is given by 2ⁿ-1. 

Consider an example, If set A has the elements, A = {a, b}, then the proper subset of the given subset are { }, {a}, and {b}.

Here, the number of elements in the set is 2. 

We know that the formula to calculate the number of proper subsets is 2ⁿ – 1. 

= 2² – 1

= 4 – 1

= 3

Thus, the number of proper subset for the given set is 3 ({ }, {a}, {b}).

What is Improper Subset?

A subset which contains all the elements of the original set is called an improper subset. 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} is the improper subsets. Therefore, we can write {2,4,6} ⊆ P.

Note: The empty set is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other set.

Power Set

The 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.

Properties of Subsets

Some of the important properties of subsets are:

• Every set is considered as a subset of the given set itself. It means that X ⊂ X or Y ⊂ Y, etc
• We can say, an empty set is considered as a subset of every set. 
• X is a subset of Y. It means that X is contained in Y
• If a set X is a subset of set Y, we can say that Y is a superset of X

Video Lesson on What are Sets

Also, read:

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.

Example 3: Find the number of subsets and the number of proper subsets for the given set A = {5, 6, 7, 8}.

Solution:

Given: A = {5, 6, 7, 8}

The number of elements in the set is 4

We know that,

The formula to calculate the number of subsets of a given set is 2ⁿ

= 2⁴ = 16

Number of subsets is 16

The formula to calculate the number of proper subsets of a given set is 2ⁿ – 1

= 2⁴ – 1

= 16 – 1 = 15

The number of proper subsets is 15.

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