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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