Power Set

Definition of power set:

We have defined a set as a collection of its elements so, if S is a set then the collection or family of all subsets of S is called the power set of S and it is denoted by P(S).

Thus, if S = a, b then the power set of S is given by P(S) = {{a}, {b}, {a, b}, ∅}

We have defined a set as a collection of its elements if the element be sets themselves, then we have a family of set or set of sets.

Thus, A = {{1}, {1, 2, 3}, {2}, {1, 2}} is a family of sets.

The null set or empty set having no element of its own is an element of the power set; since, it is a subset of all sets. The set being a subset of itself is also as an element of the power set.

For example:

1. The collection of all subsets of a non-empty set S is a set of sets. Thus, the power set of a given set is always non-empty. This set is said to be the power set of S and is denoted by P(S). If S contains N elements, then P(S) contains 2^n subsets, because a subset of P(S) is either ∅ or a subset containing r elements of S, r = 1, 2, 3, ……..

Let S = {1, 2, 3} then the power set of S is given by P(S) = {{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, ∅, S}.

2. If S = (a), then P(S) = {(a), ∅}; if again S = ∅, then P(S) = {∅}. It should be notated that ∅ ≠ {∅}. If S = {1, 2, 3} then the subset of S {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅.

Hence, P(S) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅}.

3. We know, since a set formed of all the subset of a set M as its elements is called a power set of M and is symbolically denoted by P(M). So, if M is a void set ∅, then P(M) has just one element ∅ then the power set of M is given by P(M) = {∅}

Set Theory

Sets Theory

Representation of a Set

Types of Sets

Finite Sets and Infinite Sets

Power Set

Problems on Union of Sets

Problems on Intersection of Sets

Difference of two Sets

Complement of a Set

Problems on Complement of a Set

Problems on Operation on Sets

Word Problems on Sets

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Union of Sets using Venn Diagram

Intersection of Sets using Venn Diagram

Disjoint of Sets using Venn Diagram

Difference of Sets using Venn Diagram

Examples on Venn Diagram

8th Grade Math Practice

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