# Relation

https://arbital.com/p/relation_mathematics

by Kevin Clancy May 17 2016 updated Jul 7 2016

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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.
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[summary: A relation is a set of [tuple_mathematics tuples], all of which have the same [tuple_arity arity]. 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 $R$.]

A relation is a set of [tuple_mathematics tuples], all of which have the same [todo: generalize the functionarity page to include general arity][tuplearity arity]. 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 $R$.