Relation (mathematics) Totally Explained

Everything about Mathematical Relation totally explained

ť This article sets out the set-theoretic notion of relation. For a more elementary point of view, see binary relations and triadic relations.:For a more combinatorial viewpoint, see theory of relations.

In mathematics, especially set theory, and logic, a relation is a property that assigns truth values to combinations (k-tuples) of k individuals. Typically, the property describes a possible connection between the components of a k-tuple. For a given set of k-tuples, a truth value is assigned to each k-tuple according to whether the property does or doesn't hold.
   An example of a ternary or triadic relation (for example, between three individuals) is: "X was-introduced-to Y by Z", where (X,Y,Z) is a 3-tuple of persons; for example, "Beatrice Wood was-introduced-to Henri-Pierre Roché by Marcel Duchamp" is true, while "Karl Marx was-introduced-to Friedrich Engels by Queen Victoria" is false.
   The variable k giving the number of "places" in the relation, 3 for the above example, is a non-negative integer (zero, one, two, ...), called the relation's arity, adicity, or dimension. A relation with k places is variously called a k-ary, a k-adic, or a k-dimensional relation. Relations with a finite number of places are called finite-place or finitary relations. It is possible to generalize the concept to include infinitary relations between infinitudes of individuals, for example infinite sequences; however, in this article only finitary relations are discussed, which will from now on simply be called relations.
   Since there's only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations, one for the property "is a 0-tuple", and one for its negation ("is not a 0-tuple"). One-place relations are called unary relations. For instance, any set (such as the collection of Nobel laureates) can be viewed as a collection of individuals having some property (such as that of having been awarded the Nobel prize). Two-place relations are called binary relations or dyadic relations. The latter term has historic priority. Binary relations are very common, given the ubiquity of relations such as:

Equality and inequality, denoted by signs such as "=" and "<" in statements like "5 < 12";
Being a divisor of, denoted by the sign "|" in statements like "13 | 1001";
Set membership, denoted by the sign "∈" in statements like "1 ∈ N". A k-ary relation, k ≠ 2, is a straightforward generalization of a binary relation.

Informal introduction

Relation is formally defined in the next section. In this section we introduce the concept of a relation with a familiar everyday example. Consider the relation involving three roles that people might play, expressed in a statement of the form "X thinks that Y likes Z ". The facts of a concrete situation could be organized in a Table like the following:
Relation S : X thinks that Y likes Z>

Person X	Person Y	Person Z
Alice	Bob	Denise
Charles	Alice	Bob
Charles	Charles	Alice
Denise	Denise	Denise

X thinks that Y likes Z ". For instance, the first row says, in effect, "Alice thinks that Bob likes Denise". The Table represents a relation S over the set P of people under discussion:
ť P =, and we've the binary relation D on P such that the ordered pair (n, m) is in the relation D just in case n|m. In other turns of phrase that are frequently used, one says that the number n is related by D to the number m just in case n is a factor of m, that is, just in case n divides m with no remainder. The relation D, regarded as a set of ordered pairs, consists of all pairs of numbers (n, m) such that n divides m.
For example, 2 is a factor of 4, and 6 is a factor of 72, which can be written either as 2|4 and 6|72 or as D(2, 4) and D(6, 72).

Coplanarity

For lines L in three-dimensional space, there's a ternary relation picking out the triples of lines that are coplanar. This does not reduce to the binary symmetric relation of coplanarity of pairs of lines.
   In other words, writing P(L, M, N) when the lines L, M, and N lie in a plane, and Q(L, M) for the binary relation, it isn't true that Q(L, M), Q(M, N) and Q(N, L) together imply P(L, M, N); although the converse is certainly true (any pair out of three coplanar lines is coplanar, a fortiori). There are two geometrical reasons for this.
   In one case, for example taking the x-axis, y-axis and z-axis, the three lines are concurrent, for example intersect at a single point. In another case, L, M, and N can be the three parallel edges of an infinite triangular prism.
   What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

Suggested reading

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990). Charles Peirce restated and extended De Morgan's results. Russell (1938; 1st ed. 1903) was historically important, in that it brought together in one place many 19th century results on relations, especially orders, by Peirce, Frege, Cantor, Dedekind, and others. Russell and A. N. Whitehead made free use of these results in their epochal Principia Mathematica.
Texts on set theory typically include a chapter on relations (for example, chpt. 3 in Suppes 1972). But there's no systematic treatise on the theory of relations, even though relations can be found everywhere in mathematics, logic, and theoretical science, and are well-grounded in elementary set theory. Likewise, there's no comprehensive discussion of how algebraic structures emerge from the assumed properties of relations. These lacunae are all the more amazing given that functions are relations, and function is arguably the most ubiquitous concept in all of mathematics. Carnap (1958) is an introduction to mathematical logic that includes an unusual amount of relation theory, but employs an unconventional notation and terminology. In Lucas (1999), relations are an important part of the intersection of mathematics and philosophy. For an introduction to the closely related subject of relation algebra, see Maddux (2006).

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