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.

Proper vs. Improper Subset

Basis	Proper Subset	Improper Subset
Definition	Contains only a few or no elements of the original set.	Contains all elements of the original set.
Relation to Original Set	It is not equal to the original set.	It is always equal to the original set.
Formula	The number of proper subsets of any set is (2n – 1), where ‘n’ is the number of elements in the given set.	The number of improper subsets of any set is always 1, which is the set itself.
Symbol	Denoted by the symbol ‘⊂.’	Denoted by the symbol ‘⊆.’.
Example	Example:If B = {1, 3, 5}, its proper subsets are { }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}	Example:If B = {1, 3, 5}, its improper subset is{1, 3, 5}

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