%%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. %%

[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 function*arity page to include general arity][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$~$.