1.1 Sets:  

 Definitions and Representations

Definition: A set is a collection of distinct objects.

Representation:

Roster Method: Listing all elements within curly braces. Example: A = {1, 2, 3, 4}

Set-Builder Notation: Defining a set by a property that its elements must satisfy. Example: B = {x | x is an even integer}

Types of Sets

Finite Set: A set with a finite number of elements. Example: {a, b, c}

Infinite Set: A set with an infinite number of elements. Example: {1, 2, 3, …}

Empty Set (∅ or {}): A set with no elements.

Singleton Set: A set with only one element. Example: {5}

Universal Set (U): The set of all possible elements under consideration.

Set Operations

Union (∪): The union of two sets A and B is the set of all elements that are in A, or in B, or in both.

A ∪ B = {x | x ∈ A or x ∈ B}

Intersection (∩): The intersection of two sets A and B is the set of all elements that are in both A and B.

A ∩ B = {x | x ∈ A and x ∈ B}

Complement: The complement of a set A (with respect to the universal set U) is the set of all elements in U that are not in A.

A’ = {x | x ∈ U and x ∉ A}

Difference (A – B): The difference between two sets A and B is the set of all elements that are in A but not in B.

A – B = {x | x ∈ A and x ∉ B}

Power Sets

"The power set of a set A is the set of all possible subsets of A, including the empty set and A itself. If A has n elements, then the power set of A has 2^n elements."

Example: If A = {a, b}, then the power set of A is P(A) = { {}, {a}, {b}, {a, b} }

Cartesian Products

"The Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) where a is in A and b is in B."

A × B = {(a, b) | a ∈ A and b ∈ B}

Example: If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}