Family of Sets & Power Set –
If we have a set whose elements are sets themselves, then the set formed is called the family of sets, or the class of sets or the set of sets.
For example, if A = {1, 2, 3}, B = {p, q}, C = {5, 6} then the set S whose elements are all or some of these sets is a family of sets. Then S = {{1, 2, 3}, {p, q}, {5, 6}}
The family of all subsets of a given set is called the power set. Power set of a given set A is denoted by P(A).
Thus if A = {1, 2, 3}, then its subsets are ϕ, {1}, {2}, {3}, {1, 2}, {3, 1}, {2, 3}, {1, 2, 3}
So, P(A) = {ϕ, {1}, {2}, {3}, {1, 2}, {3, 1}, {2, 3}, {1, 2, 3}}
Remarks – It may be noted that P(A) is a set. Therefore, we have {1} ∈ P(A) true,
1 ∈ P(A) false, and {1} ⊂ P(A) false. {(1)} P(A) True
Notes; -
1. ϕ and A are both elements of P(A)
2. If A = ϕ, then P(ϕ) = {ϕ}
3. If A = {a}, then P(A) = {ϕ, {a}}
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