Power Set

1.0What is a Power Set (or) Definition of Power Set

A Power Set of a set A is the set of all possible subsets of A, including the empty set and the set A itself.

If A={1,2}, then the Power Set of A, denoted by P(A), is: P(A)={ ϕ,{1},{2},{1,2} }

Thus, the power set represents the complete collection of subsets.

Notation of Power Set

• The power set of A is denoted by: P(A)
• If A has n elements, then $|P(A)|=2^n$

2.0Formula for Power Set

Number of Subsets

For a finite set A with n elements: $|P(A)|=2^n$

This is because each element of A can either be included or excluded in a subset, giving 2 choices per element.

General Expression

If $A=\{a_1,a_2,\dots ,a_n\}$

Then, P(A)={ S:S⊆A }

3.0How to Write a Power Set?

Step-by-Step Method

1. Start with the empty set (ϕ).
2. List all 1-element subsets.
3. List all 2-element subsets.
4. Continue until you list the n-element subset, which is the set A itself.

Example 1: Power Set of {1, 2}: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Power Set of {a, b, c}: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

4.0Properties of Power Set

1. If a set A has n elements, then P(A) has $2^n$ elements.
2. The empty set (ϕ) is always a member of P(A).
3. The set A itself is always a member of P(A).
4. P(A) is always larger than A, unless A is the empty set.
5. P(ϕ)={ϕ}.

5.0Examples of Power Set

Example 1: Find the power set of {1, 2}

Sol: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Find the power set of {a, b, c}

Sol: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

Example 3: If A has 5 elements, find ∣P(A)∣.

Sol: $|P(A)|=2^5=32$

6.0Applications of Power Set 

1. Relations and Functions

• A relation from set A to B is a subset of A×B.
• Understanding power sets helps in visualizing all possible relations.

2. Probability

• Sample space in probability can be considered as a power set of outcomes.
• Example: Tossing a coin twice → Sample space = {HH, HT, TH, TT}. Its power set covers all possible events.

3. Logic and Boolean Algebra: Power sets are closely related to truth tables and logical structures.
4. Combinatorics: Many combinatorial problems in JEE rely on counting subsets, which directly use the power set formula.

7.0Practice Questions on Power Set

1. Find the power set of {x, y}.
2. A set A has 6 elements. Find the number of subsets in its power set.
3. If ∣P(A)∣=64, find ∣A∣.
4. List all subsets of {1, 2, 3, 4}.
5. How many subsets of a set with 10 elements contain at least 3 elements?