# Subset

A subset is a set whose elements are all members of another set. In other words, a subset is a part of a given set.

If A and B are two sets, we say A is a subset of B if every element of A is also an element of B.

## Examples

If set A = {a, c, d, g, h} and set B = {a, b, c, d, e, f, g, h}, then set A is a subset of set B since all elements of set A are present in set B.

It is written as A ⊆ B

The line under the symbol ‘⊂’ means that set A may also be equal to set B. In such cases, they will be identical sets. However, if set A is a proper subset of set B (where at least one element of B is not in set A), we remove the line and write A ⊂ B.

Moreover, when a set is not a subset of a given set, we add a slash through the symbol ‘⊂’ and write A ⊄ B (read as A is not a subset of B).

## Proper Subset

A proper subset is a subset that contains some, but not all, elements of another set. Thus, any set is a subset of itself but not a proper subset.

If A and B are two sets, A is a proper subset of B if every element of A is in B, and A is not equal to B.

### Symbol

A proper subset is denoted by the symbol ‘⊂.’ If A is the proper subset of B, it is expressed as A ⊂ B.

### Example

If D = {a, g, h}, its proper subsets are { }, {a}, {g}, {h}, {a, g}, {a, h}, and {g, h} whereas the set {a, g, h} is not a proper subset of A.

Here is another example shown with a Venn diagram, which represents set A = {5, 8, 9} as a proper subset of set B = {1, 2, 3, 4, 5, 8, 9}:

Again, a set of whole numbers, natural numbers, integers, rational numbers, or irrational numbers are all subsets of real numbers.

Similarly, a set of real numbers is a subset of the complex numbers.

## Improper Subset

An improper subset is a subset that contains every element of another set, making it exactly equal to that set. Thus, every set has only one improper subset, the set itself.

### Symbol

An improper subset is denoted by the symbol ‘⊆.’ Set A is an improper subset of set B if A is equal to B and is written as A ⊆ B.

### Example

If D = {a, g, h}, its improper subset is {a, g, h}.

Here is another example with a Venn diagram, representing set A = {1, 2, 5, 7, 8, 9} as an improper subset of set B = {1, 2, 5, 7, 8, 9}

Now, let us summarize the differences between a proper and an improper subset.

## Number of Subsets of a Set

The number of subsets for any set with ‘n’ elements is given by the formula 2n. It includes the empty set.

Let D = {a, g, h} be a given set. It has 3 elements.

The possible subsets of D are { }, {a}, {g}, {h}, {a, g}, {a, h}, {g, h}, and {a, g, h}. It has a total of 8 subsets.

Here, instead of finding all the subsets, we will use the formula 2n = 23 = 8, which gives us the number of subsets of set D

Here are a few more examples,

• A set with 4 elements has 24 = 16 subsets
• A set with 6 elements has 26 = 64 subsets
• A set with 1 element has 21 = 2 subsets (the empty set ɸ and the original set)
• A set with 0 elements has 20 = 1 subset (the empty set ɸ)

### Formulas

• The number of proper subsets of any set with ‘n’ elements is given by the formula 2n – 1. For example, if D = {a, g, h}, it has 23 – 1 = 7 proper subsets.
• The number of improper subsets of any set with ‘n’ elements is always 1. Therefore, the number of elements in a set does not affect the number of improper subsets.

## Power Set

The power set is the collection of all the subsets of a given set, including the empty set and the original set. If A is a set with ‘n’ elements, its power set is denoted by P(A) with 2n elements.

For example,

If B = {1, 3, 5}, its P(B) = {{ }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}}

Here, we observe that B has n = 3 elements and P(B) has 2n = 23 = 8 elements.

## Solved Examples

Find the subsets, proper subsets, and their number of elements for the set A = {2, 5, 7, 12}

Solution:

Given A = {2, 5, 7, 12}
Thus, the subsets are { }, {2}, {5}, {7}, {12}, {2, 5}, {2, 7}, {2, 12}, {5, 7}, {5, 12}, {7, 12}, {2, 5, 7}, {2, 5, 12}, {2, 7, 12}, {5, 7, 12}, and {2, 5, 7, 12}
The proper subset is {2, 5, 7, 12}
Now, set A has n = 4 elements.
Thus, the subset has 2n = 24 = 16 elements, and the proper subset is always 1.

Determine whether A is a subset of B when A = {7, 9, 11}, and B = {1, 3, 5, 7, 9, 11}.

Solution:

Given A = {7, 9, 11} and B = {1, 3, 5, 7, 9, 11}
From set A,
7 ∈ B
9 ∈ B
11 ∈ B
Since all elements of set A are present in set B, A ⊆ B
Thus, A is a subset of B.