Subsets in Maths: Complete Guide for Students

In mathematics, we define the subset as showing that all the elements of some set A are contained within some other set B. There are two types of subsets, proper subsets and normal subsets.

If all elements of set A are in another set B, then set A is said to be a subset of set B.

Subsets

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 present in set “b”. In alternative words, set “a” is contained within set “b”.

Example: if set “a” has {x, y} and set “b” has {x, y, z}, then “a” is the subset of “b” as elements of “a” are present in set “b”.

Subset Symbol

A subset is denoted by the symbol ⊆.

A ⊆ b;

which means set A is a subset of set b.

All Subsets of a Set

The subsets of any set contain all possible sets, including its components 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

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 specific formula to find the subsets. Instead, we've to list them to differentiate between proper and improper ones.

Proper Subsets

Set a is considered a proper subset of set b if set b contains a minimum of one component 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 as 36 is not present within set a.

Proper Subset Symbol

A proper subset is denoted by ⊂. We will express a proper subset for set a and set b as,

A ⊂ b

Formula

If a set has “n” components, then the number of subsets of the given set is 2n, and conjointly 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 set is { }, {a}, and {b}.

Here, the number of components within 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 subsets for the given set is 3 ({ }, {a}, {b}).

What is an Improper Subset?

A subset that contains all the elements of the initial set is named an improper subset of the initial set. it's 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.

Power Set

The power set is said to be the collection of all the subsets. It's represented by p(A). If A is set having elements {a, b}.

Then the power set of “A” can be;

P(A) = {∅, {a}, {b}, {a, b}}.

Properties of Subsets

Some of the necessary properties of subsets are:

• Every set is taken into account as a subset of the given set itself. It implies that x ⊂ x or y ⊂ y, etc.

• We can say an empty set is considered a subset of each set.

• X is a subset of y. It implies that x is contained in y.

• If a set x could be a subset of set y, we can say that y may be a superset of x.

Solved Examples

1. How many subsets containing three elements can be formed from the set?

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

Ans: Number of elements within the set = 10

Number of parts within the subset = 3

Therefore, the number of possible subsets containing 3 parts= 10c3

$=\dfrac{10!}{(10-3)!.3!}$

$=\dfrac{10 \times 9 \times 8 \times 7!}{(7)!.3 \times 2 \times 1}$

$=\dfrac{720}{6}$

$=120$

Therefore, the number of possible subsets containing 3 parts from the set S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is one hundred twenty.

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

Ans: we can find a variety of examples of subsets in our way of life, such as:

1. If we consider all the books in a library as one set, then books relating to maths may be a subset.

2. If all the items in a grocery store form a collection, then cereals form a subset.

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

Ans: 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

The 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.

Conclusion

If a set has n no. of elements, then the no. of subsets of the given set is 2n, and the no. of proper subsets is 2n-1. In this article, What are a subset and its symbol, the properties of the subset, the definition of a power set and the types of subsets are explained with examples.

Practice Problems

1. Find the number of subsets and the number of proper subsets for the given set A = {5, 4, 2, 1, 0}.

2. Find the subsets for the given set A = {3, 4, 5, 6, 7}.

List of Related Articles

• Set, subset and superset

• Types of set