Definition

Noun: 1. A mathematical operation that, given two sets, returns a new set of all possible ordered pairs where the first element is from the first set and the second element is from the second set. It is a fundamental concept in set theory, used to construct relationships and multi-dimensional spaces. 2. The resulting set from this operation. This set is also called the "product set" or "direct product."

Usage and Examples

• Basic Usage: The Cartesian product of set A and set B is written as A × B.
  • Example: If A = {1, 2} and B = {x, y}, then the Cartesian product A × B = {(1, x), (1, y), (2, x), (2, y)}.
• Describing the Result: The concept is used to define coordinates and relations.
  • Example: The two-dimensional plane is the Cartesian product of the set of real numbers with itself, denoted as ℝ × ℝ or ℝ².

Advanced Usage

• n-ary Cartesian Product: The operation can be extended to more than two sets. The Cartesian product of sets A₁, A₂, ..., Aₙ is the set of all ordered n-tuples (a₁, a₂, ..., aₙ) where each aᵢ is in Aᵢ.
  • Example: In a database, a table can be viewed as a subset of the Cartesian product of its column domains.
• Empty Set: The Cartesian product of any set with the empty set is itself the empty set.
  • Example: For any set A, A × ∅ = ∅.

Variants and Related Words

• Cross Product: A common synonym for Cartesian product.
• Direct Product: Another synonym, often used in more abstract algebraic contexts.
• Ordered Pair: A fundamental component (a, b) that is an element of a Cartesian product.
• Tuple: An ordered list of elements; an element of an n-ary Cartesian product (e.g., a triple, quadruple).
• Relation: In mathematics, a relation from set A to set B is defined as a subset of the Cartesian product A × B.

Synonyms

• Cross product
• Direct product
• Product set

Related Concepts and Phrases

• Set Theory: The branch of mathematics where the Cartesian product is formally defined.
• Coordinate System: A system built using the concept of the Cartesian product, most famously the Cartesian coordinate system.
• A × B = {(a, b) | a ∈ A and b ∈ B}: The formal set-builder notation defining the Cartesian product.

Noun

1. the set of elements common to two or more sets
   • the set of red hats is the intersection of the set of hats and the set of red things