Power Set
The power set of a set $ X $ encompasses all possible subsets of that set, including the empty set and the set itself, denoted by $ P(X) $.
If a set \( A \) contains \( n \) elements, the power set of \( A \) consists of \( 2^n \) elements. This results from each element having two possible statuses: included or excluded in each subset.
The function mapping each set to the cardinality of its power set is exponential in nature, according to number theory. This illustrates the swift expansion of the power set as the original set's size increases.
Power sets play a crucial role in defining relationships and functions in mathematics, particularly in the rigorous definitions of binary relations and functions as specific subsets of Cartesian products.
Example
Consider a set with n=2 elements:
$$ A = \{a, b\} $$
The power set of \( A \) comprises four sets:
$$ P(A) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\} $$
Example 2
For another simple set with n=3 elements, consider:
$$ A = \{a, b, c\} $$
The power set $ P(A) $ is composed of eight subsets:
$$ \mathcal{P}(A) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{ab\}, \{ac\}, \{bc\}, \{a, b, c\}\} $$
With larger sets, the number of subsets grows exponentially with the size of the initial set.
Example 3
Consider a set composed of n=4 elements:
$$ A = \{a, b, c, d\} $$
The power set P(A) includes 16 subsets:
- \( \emptyset \)
- \( \{a\} \)
- \( \{b\} \)
- \( \{c\} \)
- \( \{d\} \)
- \( \{a, b\} \)
- \( \{a, c\} \)
- \( \{a, d\} \)
- \( \{b, c\} \)
- \( \{b, d\} \)
- \( \{c, d\} \)
- \( \{a, b, c\} \)
- \( \{a, b, d\} \)
- \( \{a, c, d\} \)
- \( \{b, c, d\} \)
- \( \{a, b, c, d\} \)
These subsets include all possible combinations of \( A \)'s elements, from the empty set to the full set.
