Subsets fall under the mathematics concept Sets. A Set is a collection of objects or elements enclosed within curly braces {}. If Set A is a Collection of Odd Numbers and Set B includes { 1, 3, 5} then B is said to be a subset of A and is denoted by BâŠ†A whereas A is the Superset of B.

Elements of Set can be anything such as variables, constants, real numbers, whole numbers, etc. It can include a null set too at times. Learn about Subsets Definition, Symbol, Formula, Types, and Examples in the later modules.

## What is a Subset?

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

Example: Â If Set A has {a, b} and set B has {a, b, c}, then A is the subset of B because elements of A are also present in set B.

### Subset Symbol

In Set theory, the Subset is denoted by the symbol âŠ† and is usually read as the Subset of. Using the symbol let us denote the subsets as follows

A âŠ† B; which denotes Set A is a subset of Set B.

Remember a Subset may include all the elements that are present in the Set.

Also, Read:

### All Subsets of a Set

Subsets of any set consist of all possible sets including its elements and even null set. Let us provide you an example and provide the possible combinations of a set so that you can better understand.

Example:

What are the Subsets of Set A = {5, 6, 7, 8}?

Solution:

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

Subsets include

{}

{5}, {6}, {7}, {8},

{5,6}, {5, 7}, {5,8}, {6,7},{6,8}, {7,8},

{5,6,7}, {6,7,8}, {5,7,8}, {5,6,8}

{5,6,7,8}

### Types of Subsets

Subsets are Classified into two types namely

- Proper subsets
- Improper subsets

**Proper Subsets:** Proper Set includes only a few elements of the original set.

**Improper Subsets:** Improper Set includes all the elements of the original set along with the null set.

Example:

If set A = { 4, 6, 8}, then,

Number of subsets: {4}, {6}, {8}, {4,6}, {6,8}, {4,8}, {4,6,8} and Î¦ or {}.

Proper Subsets: {}, {4}, {6}, {8}, {4,6}, {6,8}, {4,8}

Improper Subset: {4,6,8}

There is no proper formula to find how many subsets are there. You just need to write each one of them and distinguish whether it is a proper subset or improper subset.

### What are Proper Subsets?

Set A is said to be a Proper Subset of Set B if Set B has at least one element that doesn’t exist in Set A.

Example: If Set A has elements as {4, 6} and set B has elements as {4, 6, 16}, then Set A is the Proper Subset of B because 16 is not present in the set A.

### Proper Subset Symbol

A Proper Subset is represented by the symbol âŠ‚ and is read as “Proper Subset of”. Using this symbol we can denote the Proper Subset for Set A and Set B as A âŠ‚ B.

### Proper Subset Formula

If we have to pick the “n” number of elements from a set containing the “N” number of elements you can do so in ^{N}C_{n }number of ways.

### How many Subsets and Proper Subsets does a Set have?

If a Set has “n” elements then there are 2^{n} subsets for the given set and there are 2^{n} -1 proper subsets for the given set.

Consider an example, If set X has the elements, X = {x, y}, then the proper subset of the given subset are { }, {x}, and {y}.

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

We know that the formula to calculate the number of proper subsets is 2^{n}Â â€“ 1.

= 2^{2}Â â€“ 1

= 4 â€“ 1

= 3

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

### What are Improper Subsets?

A Subset that contains all the elements of the original set are called Improper Sets. It is represented using the symbol âŠ†.

Set A ={1,2,3} Then, the subsets of A are;

{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} and {1,2,3}.

Where, {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} are the proper subsets and {1,2,3} is the improper subsets. Therefore, we can write {1,2,3} âŠ† A.

Note: Empty Set is an Improper Subset of itself and is a Proper Subset of any other set.

### Power Set

Power Set is said to be a collection of all subsets. It is denoted by P(A).

If Set A has elements {a, b} then the Power Set of A is

P(A) = {âˆ…, {a}, {b}, {a, b}}

### Properties of Subsets

Below is the list of Some of the Important Properties of Subsets and they are as such

- Every Set is a subset of the given set itself. It means A âŠ‚ A or B âŠ‚ B, etc.
- Empty Set is a Subset of Every Set.
- If A is a Subset of B it means A is contained within B.
- If Set A is a subset of Set B then we can infer that B is a superset of A.

### Subsets Example Problems

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

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

Solution:

No. of Elements in the Set = 7

No. of Elements in the Subset = 3

No. of Possible Elements containing the 3 Elements is 7C_{3}

=\(\frac { 7! }{ (7-3)!*3! } \)

= \(\frac { 7*6*5*4! }{ 4!*3! } \)

= 35

2. Find the number of subsets and the number of proper subsets for the given set A = {4, 5, 6, 7,8}?

Solution:

Given: A = {4, 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 2^{n}

= 2^{5} = 32

The number of subsets is 32

The formula to calculate the number of proper subsets of a given set is 2^{n}Â â€“ 1

= 2^{5}Â â€“ 1

= 32 â€“ 1 = 31

The number of proper subsets is 31.

### FAQs on Subsets

**1. What are Types of Subsets?**

Subsets are classified into two types and they are as follows

- Proper Subsets
- Improper Subsets

**2. What is the formula to calculate the number of proper subsets and subsets for a given set?**

If a set has “n” number of elements then the number of subsets and proper subsets is given by the formulas

The formula to calculate the number of proper subsets for a given set is 2^{n}-1

The formula to find the number of subsets is 2^{n}

**3. What is the symbol to denote Subset?**

The subset is denoted by the symbol âŠ†.