# CLASS-11NUMER OF SUBSETS OF A GIVEN SETS

Number of Subsets of a Given Set

A set having ‘n’ element has 2ⁿ subsets

For Example

1. The subset of the set {1, 2} are  {1, 2}, {1}, {2}, ϕ

The number of subsets is 4, i.e.,

2. The subset of the set {1, 2, 3} are {1, 2, 3}, {2, 3}, {1, 2}, {1, 3}, {1}, {2}, {3}, ϕ

The number of subsets is 8, i.e.,

The sets of numbers N = the set of natural numbers 1, 2, 3,…………….,

W = the set of whole numbers, 0, 1, 2, 3,………….,I or

Z = the set of integers…, -2, -1, 0, 1, 2, 3,…………..,

Q = the set of rational numbers,

R = the set of real numbers.

The above sets are related as N ⊂  W ⊂  Z  ⊂  Q ⊂ R.

Example.1) If A = {x : x = 2n, n ∈ N}, and B = {x : x = 2ⁿ, n ∈ N}. Is A ⊆ B or B ⊆ A ?

Ans.)  A = {x : x = 2n, n ∈ N}

=  {2, 4, 6, 8, 10,………….}.

Here A contains all positive integral multiplies of 2

B = { x : x = 2ⁿ, n ∈ N}

=  {2, 2², 2³,…………..} = {2, 4, 8, 16,……………}

B contains all positive integral power of 2. Since a power of 2 is a multiple of 2, therefore, every element of B is in A.

So,  B ⊆ A     (Prove)

Example.2)  Prove that ϕ ⊆ A for any set A. Is it true that ϕ ⊂ A.

Ans.)  Let, if possible, ϕ be not a subset of A. Then there should be an element in ϕ which is not in A. Since there is no element in ϕ, this is not possible. Therefore, our assumption that ϕ is not a subset of A is wrong.

So,  ϕ ⊆ A

ϕ ⊆ A is not true since, if A = ϕ, then we cannot find an element of A which is not in ϕ    (Ans.)

Example.3)  Prove that, A ⊆ ϕ  =>  A = ϕ

Ans.) We know that two sets A & B are equal if A ⊆ B, and B ⊆ A

Here, we know that ϕ  ⊆  A.

Also, A ⊆ ϕ  (given)

So,  A = ϕ   (Prove)