Method to write Cartesian product of …

In <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Set_theory&hl=en-IN">set theory</a> (and, usually, in other parts of <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Mathematics&hl=en-IN">mathematics</a>), a Cartesian product is a <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Mathematical_operation&hl=en-IN">mathematical operation</a> that returns a <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Set_(mathematics)&hl=en-IN">set</a> (or product set or simply product) from multiple sets. That is, for sets Aand B, the Cartesian product A × B is the set of all <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Ordered_pair&hl=en-IN">ordered pairs</a> (a, b) where a ∈ A and b ∈ B. Products can be specified using <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Set-builder_notation&hl=en-IN">set-builder notation</a>, e.g.

{\displaystyle A\times B=\{\,(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\,\}.}

<a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Cartesian_product%23cite_note-1&hl=en-IN">[1]</a>

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).

More generally, a Cartesian product of nsets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-<a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Tuple&hl=en-IN">tuple</a>. An ordered pair is a <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Tuple%23Names_for_tuples_of_specific_lengths&hl=en-IN">2-tuple or couple</a>.

The Cartesian product is named after <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Ren%25C3%25A9_Descartes&hl=en-IN">René Descartes</a>,<a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Cartesian_product%23cite_note-2&hl=en-IN">[2]</a> whose formulation of <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Analytic_geometry&hl=en-IN">analytic geometry</a> gave rise to the concept, which is further generalized in terms of <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Direct_product&hl=en-IN">direct product</a>.

Examples

A deck of cards

<a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/File:Piatnikcards.jpg&hl=en-IN"></a>

Standard 52-card deck

An illustrative example is the <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Standard_52-card_deck&hl=en-IN">standard 52-card deck</a>. The <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Playing_cards%23Anglo-American&hl=en-IN">standard playing card</a>ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {♠, ♥, ♦, ♣} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 <a href="https://googleweblight.com/i?u=https://en.m.wikipedia.org/wiki/Ordered_pairs&hl=en-IN">ordered pairs</a>, which correspond to all 52 possible playing cards.

Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.

Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.

Both sets are distinct, even disjoint.