LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

FAMILY OF SETS AND POWER SETS

**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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