In this post we will learn about the concept of subset with properties and examples.

To fully understand the chapter, you should have basic knowledge about sets and its properties.

**What is a Subset in math?**

Suppose two sets A & B are given.

If all the elements of set B is present in set A, then set B is called subset A.

If all elements of set A is present in set B, then set A is subset of B.

Let’s try to understand the concept in simple manner.

Consider set A as a master set containing 5 elements.

A = { 3, 6, 9, 10, 12}

Now from master set A, we have copied some of its elements to form set B.

B = { 6, 10, 12 }

Since all elements of B are taken/copied from A, the set B is a subset of A.

**Representing Subset**

Subset is represented by symbol \mathtt{\subseteq }

So, if set B is subset of set A then we can write; B \mathtt{\subseteq } A

The expression B \mathtt{\subseteq } is read as ” B is subset of A ” or ” B is contained in A”

**Properties of Subset**

(01)** Every set is a subset of itself.****Explanation**:

Consider set A with following elements;

A = { 9, 7, 13, 10 }

Making a copy of set A, we get;

A’= A = { 9, 7, 13, 10 }

See that set A’ is subset of set A since all elements of A’ is present in set A.

Hence set A is subset of set A (i.e.) \mathtt{A\ \subseteq \ A} .

(02) **Empty set is a subset of every set.**

**Explanation**

Consider set A with following elements;

A = { 3, 6, 5 }

Now consider empty set B;

B = { }

Since B is an empty set with no elements, it is subset of set A or any set whatsoever.

(03)** For a set containing n elements, the total number of subset possible is given by formula** \mathtt{2^{n} \ }

**Explanation**

Consider set A with 2 elements;

A = { 14, 17 }

The number of possible subsets are:

\mathtt{\Longrightarrow \ 2^{n} \ }

Here number of elements n = 2

\mathtt{\Longrightarrow \ 2^{2}}\\\ \\ \mathtt{\Longrightarrow \ 4\ \ }

Hence, there can be 4 possible subsets for set A.

Listing all the subsets of A

(i) Empty set P = { }

(ii) Q = {14}

(iii} R = {17}

(iv) S = {14, 17}

**Types of Subset**

There are two types of subsets;

(a) Proper Subset

(b) Improper Subset

**Proper Subset**

Proper subset B contains elements of the main set A, but it should not contain all the elements so that B becomes equal to A.

Given below are** characteristics of Proper Subsets**

⟹ subsets containing all the elements of the main set are not considered as proper subset

⟹ Proper subset contain at least one element and not all the elements of the main set.

⟹ The number of proper set can be found using the formula \mathtt{2^{n} -1}

⟹ Proper Subset is denoted by symbol ” \mathtt{\subset } “

**Examples of Proper Subset**

**Example 01**

Given below is the set A with three elements.

A = { 1, 3 , 5 }

Find all the proper subsets for A

**Solution**

Given below are the proper subsets for given set A.

⟹ { }

⟹ { 1 }

⟹ { 3 }

⟹ { 5 }

⟹ { 1, 3 }

⟹ { 3, 5 }

⟹ { 1, 5 }

hence, the above 7 sets are the proper subset of A.

**Example 02**Consider the set A with four elements

A = { 10, 22, 35, 41 }

Check which of the below set B, C & D is a proper subset

B = { 35, 22, 10, 41 }

C = { 10, 41 }

D = { 6, 22 }

**Solution**

Set B is not the proper subset because it contains all the elements of set A.

Set C is the proper subset as it contains two elements of set A

D is not even a subset of A as it contains element ” 6 ” which is not present in set A.

**Improper Subset**

A **subset that contains all the elements of the master set** is called improper subset.

⟹ Improper subset is the exact replica of the given master set.

⟹ There is only one possible improper subset of a given set.

**Examples of Improper Subset**

**Example 01**

Consider the set A with 5 elements

A = { 9, 22, 73, 45, 31 }

Check which of the following below set is Improper subset

B = { 73, 22, 9 }

C = {31, 73, 22, 9, 45 }

**Solution**

Set B contains three element of set A.

Hence set B is a proper subset of A ( i.e. B ⊂ A )

Set C contains all the element of set A.

Hence it is improper subset of A.

**Subsets -Solved Problems**

(01) **Find the number of possible subsets for below set A**

A = {13, 29, 37 }

Also list all the possible subsets for the set A.

**Solution**

The number of element in set A is 3.

Formula for number of subset is \mathtt{2^{n}}

Here n = 3.

Putting the value of n in the formula.

\mathtt{\Longrightarrow \ 2^{3}}\\\ \\ \mathtt{\Longrightarrow \ 8}

Hence, there are total of 8 subsets possible for the given set.

Listing all the 8 subsets below:

⟹ { }

⟹ { 13 }

⟹ { 29 }

⟹ { 37 }

⟹ {13, 29 }

⟹ { 29, 37 }

⟹ { 13, 37 }

⟹ { 13, 29, 37 }

(02) **Given below is the set A with 6 elements**.

A = { 16, 37, 29, 82, 101, 114}

Check if the below sets are subset of A or not?

(i) B = { 37, 42 }

(ii) C = { 16, 82, 114 }

(iii) D = { }

(iv) E = {114, 29, 15}

(v) F = { 101, 37 }

**Solution**

(i) **B is not subset of A** as it contains element 42 which is not present in set A.

(iii) **C is subset of A** (C ⊆ A) as it contains all the element present in set A.

(iii) D is an empty set.

We know that **empty set is a subset of any possible set**.

Hence, D ⊆ A.

(iv) **Set E is not a subset of A **since it contain additional element 15 which is not present in set A.

(v) **F is subset of A** since both elements, 101 & 37 is present in set A.

Hence, F ⊆ A

(03) **Given below is the set A with 5 elements**

A = { 33, 21, 63, -10, 5 }

**Check if the below sets are proper or improper subsets?**

(i) B = { 5 , 33 }

(ii) C = { -10, 21, 33, 31 }

(iii) D = { 63, 21, 33, 5 }

(iv) E = { 5, 21, 63, -10, 33}

**Solution**

(i) **Set B is a proper subset of A **as it contain elements 5 & 33 which is present in set A.

(ii)** Set C is not even a subset of A**.

It contains element 31 which is not present in set A.

(iii) **Set D is a proper subset of A**.

All the element in set D is present in set A.

(iv)** Set E is a improper subset of A**.

Set contains all the elements of A. Since no element is left between set A & E, the set E is called improper subset.