1. What Is a Power Set?
A power set is the set of all possible subsets of a given set. This includes:
- the empty set
- all singleton subsets
- all larger subsets
- the set itself
If the original set is A, then its power set is written as \( P(A) \).
2. Understanding with a Simple Example
Let’s say the set is:
\( A = \{1, 2\} \)
The subsets of A are:
2.1. Listing Subsets
- \( \emptyset \)
- \{1\}
- \{2\}
- \{1,2\}
So the power set is:
\( P(A) = \{ \emptyset, \{1\}, \{2\}, \{1,2\} \} \)
3. Power Set of a 3-Element Set
For a slightly larger set:
\( B = \{a, b, c\} \)
3.1. All Subsets
Subsets include:
- Empty set: \( \emptyset \)
- Singletons: \{a\}, \{b\}, \{c\}
- Pairs: \{a,b\}, \{a,c\}, \{b,c\}
- The whole set: \{a,b,c\}
So:
\( P(B) = \{ \emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\} \} \)
4. Number of Subsets in a Power Set
If a set has n elements, then its power set has \(2^n\) subsets.
This formula includes the empty set, all smaller subsets, and the full set.
4.1. Examples
- Set with 1 element → \(2^1 = 2\) subsets
- Set with 2 elements → \(2^2 = 4\) subsets
- Set with 3 elements → \(2^3 = 8\) subsets
- Set with 4 elements → \(2^4 = 16\) subsets
5. Important Points
Some key things to remember about power sets:
5.1. Key Ideas
- The power set always contains the empty set.
- The original set is always a subset in its own power set.
- The number of subsets grows very quickly as the set gets bigger.
- For any set A, the power set P(A) is always larger than A (unless A is empty).
6. Why Power Sets Matter
Power sets are important in many mathematical areas such as probability, logic, and functions. They help explore all possible combinations of elements from a set.