##
**Subset
| Universal set**

Let's suppose, A and B are two non empty sets. The set A is said to
be the

**subset**of B if every element of A is contained in the set B. It is denoted by A ⊆ B.####
__For
example__**:**

__For example__

Let, A={a,b,c} and B={a,b,c,d,e}.

Then, A is subset
of B. A ⊆ B and read as A is subset of B or A is
contained in B. “If A is a subset of B then B is a super set of A”.

####
**Proper
Subset**

Let X be a subset of Y. Then X is said to be the proper
subset of Y if X contains at least one element less than Y.

####
__For
example__**:**

__For example__

Let, X={1,2,3} and Y={1,2,3,4,5}. Then, X is a
proper subset of Y. It is written as X ⊂ Y.

####
**Improper
subset**

Let X be subset of Y. Then X is said to be the improper
subset of Y if X and Y are equal sets. i.e. X and Y contain same elements or
every element of X is contained in Y.

####
__For
example__**:**

__For example__

Let, X = {a, e, i, o, u} and Y = {e, a, i, o, u}. Then, X is
an improper subset of Y. It is denoted by X ⊆ Y.

####
**Note:**

i. The number of subsets of a given set = 2

^{n}where n is the number of elements contained in the set.
ii. The number of a proper subset of a given set
= 2

^{n}– 1 where n is the number of elements contained in a set.
iii. An empty or null set is a subset of every
set.

###
**Universal
Set**

A set which contains all the sets under consideration as the
subsets is called a universal set. It is denoted by the capital letter U of the
English alphabet.

####
__For example__**:**

__For example__

Let, A = {a set of girl students}

B = {a set
of boy students}

C = {a set
of class 8 students}

D = {a set
of class 9 students}

Then the universal set of all these sets is U = {a set of
all students of the school}.

###
*Workout Examples*

*Workout Examples*

*Example 1: Find all the subsets of the following sets.*

*a.*

*A = {a}*

*b.*

*B = {a, b}*

*c.*

*C = {1, 2, 3}*

*Solution:*

*a. Here,*

*A = {a}*

*Subsets are: { }, {a}*

*b. Here,*

*B = {a, b}*

*Subsets are: { }, {a}, {b}, {a, b}*

*c. Here,*

*C = {1, 2, 3}*

*Subsets are: { }, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}*

*You can comment your questions or problems regarding the subsets and universal set here.*

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