Math: Cartesian Product

Definition

The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Notation

• A × B is read as "A cross B" or "the Cartesian product of A and B".
• The order of the factors matters, i.e., A × B ≠ B × A (unless A = B).

Properties

• The Cartesian product is associative, i.e., (A × B) × C = A × (B × C).
• The Cartesian product is not commutative, i.e., A × B ≠ B × A (unless A = B).
• The Cartesian product is distributive over union, i.e., A × (B ∪ C) = (A × B) ∪ (A × C).

Relationship With Relations

• A binary relation R between sets A and B can be represented as a subset of the Cartesian product A × B.
• The Cartesian product A × B represents the set of all possible pairs of elements from A and B, while a binary relation R represents a specific subset of these pairs.

Number of Elements

• If A and B are finite sets with |A| = m and |B| = n, then the Cartesian product A × B has |A × B| = m × n elements.
• In general, if A and B are infinite sets, the Cartesian product A × B is also infinite.

Definition of Cartesian Product

• The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Notation and Key Facts

• A × B is read as "A cross B" or "the Cartesian product of A and B".
• The order of the factors matters, i.e., A × B ≠ B × A (unless A = B).

Properties of Cartesian Product

• The Cartesian product is associative, i.e., (A × B) × C = A × (B × C).
• The Cartesian product is not commutative, i.e., A × B ≠ B × A (unless A = B).
• The Cartesian product is distributive over union, i.e., A × (B ∪ C) = (A × B) ∪ (A × C).

Relationship with Binary Relations

• A binary relation R between sets A and B can be represented as a subset of the Cartesian product A × B.
• The Cartesian product A × B represents the set of all possible pairs of elements from A and B, while a binary relation R represents a specific subset of these pairs.

Number of Elements in Cartesian Product

• If A and B are finite sets with |A| = m and |B| = n, then the Cartesian product A × B has |A × B| = m × n elements.
• In general, if A and B are infinite sets, the Cartesian product A × B is also infinite.

Applications of Cartesian Product

• Used in database management systems to perform joins and combine data from multiple tables
• Applied in statistics to analyze and visualize multivariate data
• Utilized in computer graphics to perform transformations and projections
• Employed in cryptography to define cryptographic protocols

Cartesian Product and Relations

• A binary relation R between sets A and B is a subset of the Cartesian product A × B
• The Cartesian product A × B represents all possible pairs of elements from A and B
• Relations are defined as subsets of the Cartesian product, and operations on relations can be expressed using set operations on the Cartesian product

Properties of Cartesian Product

• The Cartesian product is commutative: A × B = B × A
• The Cartesian product is associative: (A × B) × C = A × (B × C)
• The Cartesian product is distributive over union: A × (B ∪ C) = (A × B) ∪ (A × C)

Number of Elements in Cartesian Product

• The number of elements in the Cartesian product A × B is the product of the number of elements in A and B: |A × B| = |A| × |B|
• This result can be generalized to the Cartesian product of n sets: |A₁ × A₂ × … × An| = |A₁| × |A₂| × … × |An|