Sets and their Representations A set is a non-empty collection of distinct objects. A set can be represented in various ways, including: Roster notation...
Sets and their Representations
A set is a non-empty collection of distinct objects. A set can be represented in various ways, including:
Roster notation: A set can be defined by a list of its elements, separated by commas. For example, the set {1, 2, 3, 4, 5} can be represented as {1, 2, 3, 4, 5}.
Set builder notation: A set can be defined using a set builder. A set builder is a formal expression that describes the set using logical operators. For example, the set of all natural numbers less than 10 can be described by the set builder {x | x is a natural number and x < 10}.
Geometric representation: A set can be represented visually using geometric shapes, such as circles, rectangles, or triangles. For example, the set of all points in the first quadrant can be represented by the circle centered at the origin with radius 1.
Power Set
The power set of a set S is the set of all subsets of S. In other words, it is the set of all sets that can be formed by taking any subset of S. The power set of S is denoted by P(S).
For example, if S = {1, 2, 3}, then the power set of S is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
The power set of a set can be found by using a combination of set theory and set builder notation. For example, the power set of S can be found by listing all the subsets of S and then taking the union of all these subsets