Properties of Power Set

The power set of a set A, denoted as P(A), is the collection of all subsets of A, including:

• The empty set {}
• All single-element subsets {a}
• All possible combinations of elements from A, up to the entire set A itself.

For example, if A={1,2}, the power set P(A) is {{},{1},{2},{1,2}}.

Properties of Power Sets

Some of the common properties of Power Sets are:

• The empty set {} or ϕ is always a subset of every set, and hence it is always an element of the power set.

• The power set of the empty set {} is {{}}, containing only one subset—the empty set itself.
  • S = {} ⇒ P(S) = {{}}

• Every element of the power set P(S) is a subset of S.

• The union and intersection of subsets in the power set P(S) also belong to P(S). This is because the union or intersection of any subsets of S is itself a subset of S.

• The union of all elements in the power set is the original set A, and the intersection of all elements is the empty set.

• If S is an infinite set, the power set P(S) has a strictly greater cardinality than S. For example:
  • If S is a countably infinite set, P(S) is uncountably infinite.

• The power set is closed under operations like:
  • Union: A ∪ B ∈ P(S) for all A,B ∈ P(S).
  • Intersection: A ∩ B ∈ P(S) for all A, B ∈ P(S).
  • Complement: If S is a universal set, the complement of any subset in P(S) is also in P(S).

• Each subset A ∈ P(S) has a complement subset S∖A in P(S). This duality is central in set theory.

Number of Elements in Power Set

For a given set S with n elements, number of elements in P(S) is 2ⁿ. As each element has two possibilities (present or absent}, possible subsets are 2×2×2.. n times = 2ⁿ. Therefore, the power set contains 2^n elements. 

Note:

• The power set of a finite set is finite.
• Set S is an element of the power set of S which can be written as S ϵ P(S).
• Empty Set ɸ is an element of the power set of S which can be written as ɸ ϵ P(S).
• Empty set ɸ is a subset of the power set of S which can be written as ɸ ⊂ P(S).

Read More,

• Empty Set
• Subset
• Union of Sets
• Intersection of Sets
• Cardinality of Set

Solved Examples of Power Sets

Example 1: Let A = {a, b}. The power set P(A) is:

P(A) = {{}, {a}, {b}, {a, b}}

There are 2² =4 subsets

Example 2: For B = {1, 2, 3}, the power set P(B) is:

P(B) = {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

There are 2³ = 8 subsets.

Example 3: If S = {1, 2, 3, 4}, find the cardinality of P(S)P(S)P(S).

The number of elements in S is ∣S∣ = 4.

The cardinality of P(S) is ∣P(S)∣ = 2^∣S∣ = 2⁴ = 16.

Thus, P(S) has 16 subsets.