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Power set
Power set: A power set is a collection of all subsets of a given set. In simpler terms, it's the set of all sets that can be formed by picking elements from...
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Power set: A power set is a collection of all subsets of a given set. In simpler terms, it's the set of all sets that can be formed by picking elements from...
Power set: A power set is a collection of all subsets of a given set. In simpler terms, it's the set of all sets that can be formed by picking elements from the original set.
Notation:
A set S can be represented as {a, b, c...}.
The power set of S, denoted by 2^S, is the set of all subsets of S.
Each element in 2^S is a set, and the elements are subsets of S.
Example: Let S = {1, 2, 3}.
The subsets of S are: {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
Therefore, 2^S = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
Key properties of power sets:
The power set of an empty set is the empty set.
The power set of a set with n elements has 2^n elements.
The power set of a set S is a subset of the power set of S.
Power sets are used in various mathematical problems, including determining the number of subsets of a set, finding the largest element, and solving optimization problems. They also have applications in computer science, where they are used to model the relationships between different sets of data