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Method to write Cartesian product of two sets. And please give me a example of the question.\xa0 |
| Answer» In\xa0set theory\xa0(and, usually, in other parts of\xa0mathematics), a\xa0Cartesian product\xa0is a\xa0mathematical operation\xa0that returns a\xa0set\xa0(or\xa0product set\xa0or simply\xa0product) from multiple sets. That is, for sets\xa0Aand\xa0B, the Cartesian product\xa0A\xa0×\xa0B\xa0is the set of all\xa0ordered pairs\xa0(a,\xa0b)\xa0where\xa0a\xa0∈\xa0A\xa0and\xa0b\xa0∈\xa0B. Products can be specified using\xa0set-builder notation, e.g.{\\displaystyle A\\times B=\\{\\,(a,b)\\mid a\\in A\\ {\\mbox{ and }}\\ b\\in B\\,\\}.}[1]A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product\xa0rows\xa0×\xa0columns\xa0is taken, the cells of the table contain ordered pairs of the form\xa0(row value, column value).More generally, a Cartesian product of\xa0nsets, also known as an\xa0n-fold Cartesian product, can be represented by an array of\xa0n\xa0dimensions, where each element is an\xa0n-tuple. An ordered pair is a\xa02-tuple or couple.The Cartesian product is named after\xa0René Descartes,[2]\xa0whose formulation of\xa0analytic geometry\xa0gave rise to the concept, which is further generalized in terms of\xa0direct product.ExamplesA deck of cardsStandard 52-card deckAn illustrative example is the\xa0standard 52-card deck. The\xa0standard playing cardranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits\xa0{♠,\xa0♥,\xa0♦, ♣}\xa0form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52\xa0ordered pairs, which correspond to all 52 possible playing cards.Ranks\xa0×\xa0Suits\xa0returns a set of the form {(A, ♠), (A,\xa0♥), (A,\xa0♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2,\xa0♥), (2,\xa0♦), (2, ♣)}.Suits\xa0×\xa0Ranks\xa0returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.Both sets are distinct, even disjoint. | |
