LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

NUMER 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., 2²**

**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., 2³**

**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)**

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