Relation

com­ment: I do not want to be short­ened. The mo­ti­va­tion for this is that I would pre­fer that some­one has the abil­ity to learn ev­ery­thing they need to know about re­la­tions just by read­ing the popup sum­mary.

A re­la­tion is a set of tu­ples, all of which have the same gen­er­al­ize the func­tion_ar­ity page to in­clude gen­eral ar­it­yarity. The in­clu­sion of a tu­ple in a re­la­tion in­di­cates that the com­po­nents of the tu­ple are re­lated. A set of \(n\)-tu­ples is called an \(n\)-ary re­la­tion. Sets of pairs are called bi­nary re­la­tions, sets of triples are called ternary re­la­tions, etc.

Ex­am­ples of bi­nary re­la­tions in­clude the equal­ity re­la­tion on nat­u­ral num­bers \(\{ (0,0), (1,1), (2,2), ... \}\) and the pre­de­ces­sor re­la­tion \(\{ (0,1), (1,2), (2,3), ... \}\). When a sym­bol is used to de­note a spe­cific bi­nary re­la­tion (\(R\) is com­monly used for this pur­pose), that sym­bol can be used with in­fix no­ta­tion to de­note set mem­ber­ship: \(xRy\) means that the pair \((x,y)\) is an el­e­ment of the set \(R\).

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  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.