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Cartesian product
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Definition
Definition and Notation
In Set theory | set theory, the Cartesian product of two Set (mathematics) | sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. In other words, it's a way to combine the elements of two sets to create pairs where the order of the elements matters. For example, if A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
Applications and Generalizations
The Cartesian product is a fundamental concept in mathematics with applications in various fields. It's used in Relational database | relational databases to represent relationships between entities, in Computer science | computer science for defining data structures, and in Geometry | geometry for describing coordinates in multi-dimensional spaces. The concept can be generalized to the Cartesian product of more than two sets, where the result is a set of ordered tuples. For instance, the Cartesian product of three sets A, B, and C would be A × B × C, containing all ordered triples (a, b, c) where a is in A, b is in B, and c is in C.
- Snippet from Wikipedia: Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Detailed Summary
Definition and Notation
In Set theory | set theory, the Cartesian product of two Set (mathematics) | sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. In simpler terms, it combines every element from one set with every element from the other, forming unique pairs.
Historical Origin
The concept of the Cartesian product is named after René Descartes, a France | French philosopher and mathematician of the 17th century | 17th-century. His work in Analytic geometry | analytic geometry laid the foundation for the development of the Cartesian coordinate system and, by extension, the notion of the Cartesian product.
Formal Representation
Formally, the Cartesian product of two sets A and B is defined as:
A × B = {(a, b) ]] | * × represents the Cartesian product operation. * (a, b) represents an ordered pair. * ∈ denotes set membership (e.g., a ∈ A means "a is an element of A"). == Python Implementation == In [[Python programming language | Python, the Itertools module provides the `product` function for generating Cartesian products:
```python from itertools import product
A = [1, 2] B = ['x', 'y']
cartesian_product = list(product(A, B)) print(cartesian_product) # Output: [(1, 'x'), (1, 'y'), (2, 'x'), (2, 'y')] ```
Cardinality
The cardinality (number of elements) of the Cartesian product of two finite sets is the product of their individual cardinalities. If A has m elements and B has n elements, then A × B has m * n elements.
Generalization to Multiple Sets
The Cartesian product can be extended to more than two sets. The Cartesian product of n sets A1, A2, …, An is denoted as A1 × A2 × … × An and consists of all ordered n-tuples (a1, a2, …, an) where ai ∈ Ai for each i.
Applications in Mathematics
The Cartesian product is fundamental in various mathematical fields. It is used to define:
- Relations (mathematics) | Relations: A binary relation between sets A and B is a subset of A × B.
- Function (mathematics) | Functions: A function from A to B can be viewed as a specific type of relation in A × B.
- Metric space | Metric spaces: Cartesian products are used to construct metric spaces, which are fundamental in analysis and topology.
Applications in Computer Science
In Computer science | computer science, the Cartesian product is used in:
- Relational database | Databases: To define relationships between tables.
- Programming language | Programming languages: In the form of tuples and records to represent structured data.
- Combinatorics: To enumerate all possible combinations of elements from different sets.
Applications in Other Fields
The concept of Cartesian product also finds applications in:
- Physics: In describing the state space of a system with multiple independent variables.
- Economics: In analyzing the possible outcomes of multiple economic choices.
- Linguistics: In studying the combinations of words and phrases to form sentences.
Conclusion
The Cartesian product is a powerful and versatile concept with widespread applications in various fields. It provides a systematic way to combine elements from different sets, enabling the exploration of relationships, structures, and possibilities in a wide range of contexts. (GgGm)
Cartesian product Glossary
Return to Cartesian product, Cartesian product Glossary
Ordered Pair (Couple Ordonné) A pair of elements (a, b) where the order in which the elements appear is significant.
Set (mathematics) | Set (Ensemble) A well-defined collection of distinct objects, considered as an object in its own right.
Element (mathematics) | Element (Élément) An object that is a member of a set.
Cartesian Product (Produit Cartésien) The set of all ordered pairs (a, b), where a is in A and b is in B, given two sets A and B.
Tuple A finite ordered list (sequence) of elements.
Cardinality The number of elements in a finite set.
Relation (mathematics) | Relation A subset of the Cartesian product of two sets. It defines a relationship between elements of the sets.
Function (mathematics) | Function A specific type of relation where each element in the domain is associated with exactly one element in the codomain.
Domain of a function | Domain (Domaine) The set of all possible input values (first coordinates) of a function.
Codomain The set of all possible output values (second coordinates) of a function.
Binary Relation (Relation binaire) A relation that associates elements from one set to elements in another set.
Arity (Arité) The number of arguments or operands taken by a function or operation.
Ordered n-tuple (n-uplet ordonné) A generalization of an ordered pair to an ordered list of n elements.
Power Set (Ensemble des parties) The set of all subsets of a given set.
Projection (set theory) | Projection An operation that creates a new set by selecting specific components from the elements of an existing set.
Mapping (Application (mathématiques) | Application) A synonym for “function,” emphasizing the idea of associating elements between sets.
Cross Product (Produit vectoriel) In Linear Algebra | linear algebra, the cross product is a binary operation on two vectors in three-dimensional space that results in another vector perpendicular to both.
Direct Product (Produit direct) A generalization of the Cartesian product to algebraic structures like groups or vector spaces.
Disjoint Sets (Ensembles disjoints) Sets that have no elements in common.
Intersection (set theory) | Intersection (Intersection ensembliste) The set of elements that are common to two or more sets.
Empty Set (Ensemble vide) A set that contains no elements.
Finite Set (Ensemble fini) A set that has a finite number of elements.
Infinite Set (Ensemble infini) A set that has an infinite number of elements.
Subset (Sous-ensemble) A set A is a subset of set B if all elements of A are also elements of B.
Superset (Sur-ensemble) A set B is a superset of set A if all elements of A are also elements of B.
Union (set theory) | Union (Union (mathématiques) | Union) The set of all elements that are in either of two or more sets.
Relative Complement (Complémentaire (théorie des ensembles) | Complémentaire) The set of elements in one set that are not in another set.
Symmetric Difference (Différence symétrique) The set of elements that are in either of two sets, but not in both.
Ordered Pair Notation The standard way of writing ordered pairs using parentheses and commas, e.g., (a, b).
Venn Diagram (Diagramme de Venn) A graphical representation of sets and their relationships, using overlapping circles or other shapes.
Equality (mathematics) | Equality of sets Two sets are equal if and only if they have exactly the same elements.
Set-builder notation (compréhension (mathématiques) | compréhension) A notation for defining a set by describing the properties that its elements must satisfy.
Family of sets (Famille d'ensembles) A collection of sets, often indexed by another set.
Partition of a set (Partition d'un ensemble) A division of a set into non-empty subsets such that each element belongs to exactly one subset.
Cartesian Square (Carré cartésien) The Cartesian product of a set with itself (A × A).
n-ary Cartesian Product (Produit cartésien n-aire) The Cartesian product of n sets.
Countable set (Ensemble dénombrable) A set that can be put into a one-to-one correspondence with the set of natural numbers.
Uncountable set (Ensemble non dénombrable) A set that is not countable.
Axiom of Choice (Axiome du choix) An axiom of set theory that allows for the selection of one element from each set in a collection of non-empty sets.
Continuum Hypothesis (Hypothèse du continu) A hypothesis in set theory that states there is no set whose cardinality is strictly between that of the integers and the real numbers.
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