Relation (mathematics)

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In mathematics, an n-ary relation on n sets, is any subset of
Cartesian product of the n sets
(i.e., a collection of n-tuples),[1] with the most common one being a binary relation,
a collection of order pairs from two sets containing an object from each set.[2]

The relation is homogeneous when it is formed with one set.

The homogeneous binary relations are studied for properties like reflexiveness,
symmetry, and transitivity, which determine different kinds of orderings on the set.[3]

Heterogeneous n-ary relations are used in the semantics of predicate calculus,
and in relational databases.

https://simple.wikipedia.org/wiki/Relation_(mathematics)

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In mathematics, a tuple is a finite ordered list (sequence) of elements.
An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer.

There is only one 0-tuple, referred to as the empty tuple.
An n-tuple is defined inductively using the construction of an ordered pair.

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Binary relation

In mathematics (specifically set theory),
a binary relation over sets X and Y is
a subset of the Cartesian product X × Y; that is,
it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y.[1]

It encodes the common concept of relation:
an element x is related to an element y, if and only if the pair (x, y)
belongs to the set of ordered pairs that defines the binary relation.

A binary relation is the most studied special case n = 2 of
an n-ary relation over sets X1, ..., Xn,
which is a subset of the Cartesian product X1 × ... × Xn.[1][2]