What are Power Sets?

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How To Define Power Set?

The Power set of Set S is defined in set theory as the set of all subsets of Set S, including the set itself, and the null or empty set. 

The Power set is indicated by P (S). 

Basically, this set is a combination of all subsets of a given set, including the null set.

Ex: If S=1, 2, 3 then the Power Set of a Set S is P(S) = {{1}, {2}, {3}, {1,2}, {2,3}, {3,1}, {1, 2, 3}, {}}

Where {1}, {2}, {3}, {1,2}, {2,3}, {3,1} are subsets.

{1,2,3} is set S

{} or ∅ (phi) is the Null or Empty set.


Power Set Generator

If the given set has n members, then the Power set will have 2n members.

For example 

If set A has 2 natural numbers {1, 2}, then the Power set of Natural numbers will be 22=4 elements.

So, the power set will have {1}, {2}, {1, 2} and a Null set.

P(A) = {{1}, {2}, {1, 2}, {}}

Similarly, if the Set B has 3 natural numbers say 1/2, 1/4, 1/6. Then the Power set of Real numbers will be 23=8 elements. 

P(B)= {{1/2}, {1/4}, {1/6}, {1/2, 1/4}, {1/4 , 1/6}, {1/6, 1/2}, {1/2, 1/4, 1/6}, {}}

From this, we can conclude that the Power set will always be larger than the original set. Also, the power set elements denote the number of subsets in the Power set of a Power set.


Properties of Power Set

  • Cantor's diagonal argument proves that the Power set of a set is much bigger than the original set.

  • In the Power set of A, if the number of elements is 2n, then n is the number of elements in the original A.

  • Cantor's theorem shows that the power set is finite i.e it is countable.

  • For a set of natural numbers, we can map the resulting Power set, P(S), with the real numbers. 

  • When operating with the union of sets, the intersection of sets and the complement of sets, Power set, P(S) of set S denotes the Boolean Algebra.

Power Set of Null Set

This set is also called as “Power set of empty set” or “Power set of Phi (∅)”. The Power set of a Null set is Zero.


Properties of Null set:

  • There are zero elements in a Null set. 

  • It is one of the subsets in the Power set.

  • It is represented by {} or ∅.

Problems on Power set

1) Find the Power Set and the Total Number of Elements in the Power Set if the Set A= {3, 5, 7}.

Ans: Here the set A is having a 3 set of natural numbers. Therefore the total number of elements in the Power set is 23=8.

The subsets are as follows:

{3}

{5}

{7}

{3, 5}

{5, 7}

{7, 3}

{3, 5, 7}

Null set {} or ∅

Therefore the Power set of Natural numbers will be P(A) = {{3}, {5}, {7}, {3, 5}, {5, 7}, {7, 3}, {3, 5, 7}, {}}


2) In a Cage, there are Four Sets of Animals: a Dog, Cat, Tiger and Mice. Find the Total Number of Animals in the Power Set and Also Find the Power Set of these Animals.

Ans: Here we have 4 animals, let us assign Dog as D, Cat as C, Tiger as T and Mice as M. There the set C= {D, C, T, M}. So the total number of animals in the Power set of C will be 24=16.

The subsets of the animals are as follows:

{D}

{C}

{T}

{M}

{D, C}

{D, T}

{D, M}

{C, T}

{C, M}

{T, M}

{D, C, T}

{D, C, M}

{D, T, M}

{C, T, M}

{D, C, T, M}

Null set {} or ∅

Therefore the Power set of the animals will be 

P(C) = {{D}, {C}, {T}, {M}, {D, C}, {D, T}, {D, M}, {C, T}, {C, M}, {T, M}, {D, C, T}, {D, C, M}, {D, T, M}, {C, T, M}, {D, C, T, M}, {}}


Conclusion

Students will ask why we need a power set. The key thing that a power set is useful for is to have a world to take place in other things. A lot of mathematics fields begin by defining certain subsets of a set as unique. Topology deals with open sets, measure theory deals with measurable sets, a non-principle ultrafilter requires even non-standard analysis to get going. But a Power set puts all these things under a single set.

FAQs (Frequently Asked Questions)

1. Define Power Set.

Ans: We have defined a set as a collection of its components, so if S is a set, the collection or family of all subsets of S is called the power set of S which is denoted by P(S).

2. What is the Power Set of Phi?

Ans: Set of Phi is an empty or null set. Therefore the Power set of Phi denotes a Null set or an Empty set which is denoted by {} or ∅.

3. How to Find the Elements in the Power Set?

Ans: If the original set has ‘n’ elements then the power set will have 2n elements which also gives us the number of subsets in the Power set of a set.

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