Consider the two sets

*A*= {5, 4, 8} and*B*= {1, 5, 6, 4, 8}.
Do you notice any relation between sets

*A*and*B*?
We can observe that every element of

*A*is an element of*B*. In this case, we call*A*to be a subset of*B*. Mathematically, we write it as*A*⊆*B*.If A and B are any two sets, then set A is said to be a subset of set B if every element of A is also an element of B. We write it as A ⊆ B (read as ‘A is a subset of B’ or ‘A is contained inB’). |

So, from the definition, we conclude that:

**A****⊆**

**B****if and only if**

**x****∈**

**A****implies**

**x****∈**

**B.**
Now, if

*A*⊆*B*, we can also say that*B*contains*A*. In this case, we say that*B*is a**superset**of*A.*We write it as*B ⊇**A*(read as ‘*B*contains*A*’ or ‘*B*is a superset of*A*’).
Now, consider the two sets

*A*= {letters of FOLLOW} and*B*= {letters of LOWER}.
Then,

*A*= {F, O, L, W} and*B*= {L, O, W, E, R}**Is set**

**A****a subset of set**

**B****?**

Observe that there is an element,

*F*in set*A*which is not a member of set*B*.
So,

*A*is not a subset of*B*. We write ‘*A*is not a subset of*B*’ as ‘*A B*’. This is also read as ‘*A*is not contained in*B*’.**If there exists at least one element in**

**A****which is not an element of**

**B****, then**

**A****is not a subset of**

**B****. Mathematically, we write it as**

**A**

**B****.**

Another concept you need to know about is that of proper subsets.

To understand what we mean by a proper subset, let us look at the two sets given below.

*A*= {4, 8, 12}

*B*= {2, 4, 6, 8, 10, 12, 14}

Observe that

*A*is a subset of*B*.**Is there any element in****B****which is not a member of****A****?**
Yes, 2, 6, 10, 14 ∈

*B*; however, 2, 6, 10, 14 ∉*A*.
So, in this case, we say that

*A*is a**proper subset**of*B*.Let A be any set and B be a non-empty set. Set A is called a proper subset of B if and only if every member of A is also a member of B, and there exists at least one element in B which is not a member of A. We write it as A ⊂ B. |

In simple language, we say that

*A*is a proper subset of*B*, i.e.,*A*⊂*B*, if*A*is a subset of*B*and*A ≠**B*.
Conversely, if two sets

*A*and*B*are such that*A*⊂*B*and*A*≠*B*, then*A*is called a**proper subset****of***B*and*B*is called the**superset****of***A*.
The following are some important points to be noted.

**(a) Every set is a subset of itself.**

**(b) A subset which is not a proper subset is called an improper subset. If**

**A****and**

**B****are two equal sets, then**

**A****and**

**B****are improper subsets of each other.**

**(c) Every set has only one improper subset and that is itself.**

**(d) An empty set is a subset of every set.**

**(e) An empty set is a proper subset of every set except itself.**

**(f) Every set is a subset of the universal set.**

**(g) If**

**X****⊆**

**Y****and**

**Y****⊆**

**X****, then**

**X****=**

**Y**
Now, when given any finite set

*A*, we know how to find its cardinal number.**Can we also find the number of subsets and the number of proper subsets of set****A****?**
Yes, we can.

To know what these numbers are, let us suppose that the cardinal number of the set

*A*is*m*, i.e.,*n*(*A*) =*m*, then**The number of subsets of**

**A****= 2**

^{m}

**The number of proper subsets of**

**A****= 2**

^{m }

**–****1**

For example:

If

*A*= {1, 3, 5}, then*n*(*A*) = 3.
∴ Number of subsets of

*A*= 2^{3}= 8
These are: Φ, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}

Number of proper subsets of

*A*= 2^{3 }– 1 = 8 – 1 = 7
Now, let us discuss about

**power set.**
The collection of all subsets of a set

*A*is called the**power set**of*A*. It is denoted by P(*A*). In P(*A*), every element is a set.
If the number of elements in set

i.e.,

*A*is*m*, then the number of elements in the power set of*A*is 2^{m}.i.e.,

*n*P(*A*) = 2^{m}, where*n*(*A*) =*m*
We have another type of a set called

**universal set**and it can be defined as follows:A set that contains all the elements under consideration in a given problem is called universal set and it is denoted by U or S or ξ. |

**Let us consider three sets**

**X****,**

**Y****, and**

**U****as:**

*X*= {the consonants of English alphabets}

*Y*= {all the letters of the word EDUCATION}

*U*= {all the letters in English alphabet}

**Did you notice any relation between the sets**

**X****,**

**Y****, and**

**U****?**

Let us see.

We have,

*X*= {B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z},*Y*= {E, D, U, C, A, T, I, O, N} and*U*= {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}
Observe that the set

*U*contains all the elements of both the sets*X*and*Y*. In this, we say that set*U*is a**universal set**for the sets*X*and*Y***.**
Universal set may vary from problem to problem.

For example, if we consider a set as {0, 1}, then we may consider its universal set as

**W**or**Z**or {–1, 0, 1, 2} etc. Similarly, if we consider the set as {Ganga, Yamuna, Saraswati}, then we may consider the universal set as {the rivers in India}.
Therefore, we should always specify the universal set for a given problem.

Now, let us learn to represent the information related to above discussed concepts using Venn diagrams.

Look at the following diagram.

Here,

*U*,*A*,*B*and*C*are four sets.
It can be seen that A is the subset of B i.e.,

*A***⊆***B.*
Also, set

*A*is completely contained by set*B*i.e., set*B*⊂ set*A*, we can say that*A*?*B*.
Similarly, set

*C*is neither contained by set*B*nor by set*A*. Thus,*C**B*and*C**A*.
Also, it can be seen that set

*U*contains all of the sets*A*,*B*and*C*. Thus,*U*is universal set.
Let’s now look at some examples to improve our understanding of the above discussed concepts.

**Example 1:**

**With respect to the three sets:**

**A****= {5, 10, 15, 20},**

**B****= {1, 2, 3, …, 20}, and**

**C****= {2, 4, 6, 8, 10, 12, 16, 18, 20}, classify the following statements as true or false?**

(a) A⊂B

(b) B⊆ C

(c) A ⊆ C

(d) Φ⊆A

(e) B⊆ U

**Solution:**

(a) True

Since 5, 10, 15, 20 ∈ A and 5, 10, 15, 20 ∈ B

Also, there exist many elements which are a member of set B, but not of set A.

So, A⊂ B

(b) False

Since 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 ∈ B and these elements do not belong to C

So, B

*C*
(c) False

Since 5, 15 ∈ A, but 5, 15 ∉ C

So, A

*C*
(d) True

Since an empty set is a subset of every set

So, Φ ⊆ A

(e) True.

Since every set is a subset of the universal set

So, B ⊆ U

**Example 2:**

**Write all the subsets of the set {3, 6, 9, 12}. Which of these are proper subsets and which are improper subsets?**

**Solution:**

Let

*A*= {3, 6, 9, 12}
The subsets of set

*A*are:
The proper subsets are

The improper subset is {3, 6, 9, 12}, i.e., set

*A*itself.**Example 3:**

**Three sets are defined as**

**A**

**= {1, 3, 4},**

**B**

**= {3, 4}**

**and**

**C**

**= {3, 4, 2, 1}. Prove that**

*A*and*C*are the super sets of*B*.**Solution:**

We have,

*A*= {1, 3, 4},

*B*= {3, 4} and

*C*= {3, 4, 2, 1}

It can be seen that all the elements of

i.e.,

Thus,

*B*are also in sets*A*and*C*. Also,*B*⊂*A*and*B*⊂*C*. This means that*B*is a proper subset of*A*as well as*C*.i.e.,

*B*⊂*A*and*B*⊂*C.*Thus,

*A*and*C*are the supersets of*B.***Example 4:**

**If**

**A****= {**–

**1,**–

**2,**–

**3,**–

**4,**–

**5}, then find the number of subsets of set**

**A****. Also, find the number of proper subsets of set**

**A****.**

**Solution:**

*A*= {–1, –2, –3, –4, –5},

∴

*n*(*A*) = 5 =*m*(say)
Number of subsets of

*A*= 2^{m}= 2^{5}= 32
Number of proper subsets of

*A*= 2^{m}– 1 = 32 – 1 = 31**Example 5:**

**Write the power set for the set**

**A**

**= {1, 3, 5}.**

**Solution:**

First, let us write all the subsets of set

*A*.
{Φ}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}

P(

*A*) = {{Φ}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}}**Example 6:**

**Given set**

*A*= {*x*:*x*is a natural number less than 10} and set*B*= {*y*:*y*is an even number less than 9}. Is set*B*a subset of set*A*? If yes then draw the Venn-diagram depicting the given sets.**Solution:**

Given,

*A*= {*x*:*x*is a natural number less than 10}
i.e.,

*A*= {1, 2, 3, 4, 5, 6, 7, 8, 9}*B*= {

*y*:

*y*is an even number less than 9}

i.e.,

*B*= {2, 4, 6, 8}
It is clear that

*B*⊂*A*.
Venn-diagram for

*B*⊂*A*is shown below:**Example 7:**

**Let**

*A*and*B*be two finite sets such that*n*(*A*) =*m*and*n*(*B*) =*n*. If the ratio of the number of elements of power sets of*A*and*B*is 64 and*n*(*A*) +*n*(*B*) = 32, find the value of*m*and*n*.

**Solution:***Given that*

*n(A) = m*

*n(B) = n*

The ratio of the number of elements of power sets of

*A*and*B*is 64.
∴

*m*–*n*= 6 ...(1)
Also,

*n(A) + (B)=*32
∴

*m*+*n*= 32 ...(2)
From equations (1) and (2), we get

**19**

*m*=**13**

*n*=