Binary relation Totally Explained

Everything about Binary Relation totally explained

In mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of another set. An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that's a multiple of p, but no other. In this relation, for instance, the prime 2 is associated with numbers that include -4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.
   Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, and many more. The all-important concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science, especially within the relational model for databases.
   A binary relation is the special case of an n-ary relation, that is, a set of n-tuples where the j^th component of each n-tuple is taken from the j^th domain X_j of the relation. An n-ary relation among elements of a single set is said to be homogeneous.
   In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain and codomain, respectively, of the relation, and G is called its graph.
The statement (x,y) ∈ R is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation corresponds to viewing R as the characteristic function of the set of pairs G.
The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independently of each other.

Is a relation more than its graph?

According to the definition above, two relations with the same graph may be different, if they differ in the sets X and Y. For example, if G = . The order of R and S in the notation S o R, used here agrees with the standard notational order for composition of functions.

Complement

If R is a binary relation over X and Y, then the following too:

The complement S is defined as x S y iff not x R y. The complement of the inverse is the inverse of the complement.
If X = Y the complement has the following properties:

If a relation is symmetric, the complement is too.

The complement of a reflexive relation is irreflexive and vice versa.

The complement of a strict weak order is a total preorder and vice versa. The complement of the inverse has these same properties.

Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x and y are in S.
   If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total,, a partial order, total order, strict weak order, (weak order), or an equivalence relation, its restrictions are too.
   However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, for example, in general not equal.
   Also, the various concepts of completeness (not to be confused with being "total") don't carry over to restrictions. For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers this supremum isn't necessarily rational, so the same property doesn't hold on the restriction of the relation "≤" to the set of rational numbers.

Sets versus classes

Certain mathematical "relations", such as "equal to", "member of", and "subset of", can't be understood to be binary relations as defined above, because their domains and codomains can't be taken to be sets in the usual systems of axiomatic set theory.
For example, if we try to model the general concept of "equality" as a binary relation

=

, we must take the domain and codomain to be the "set of all sets", which isn't a set in the usual set theory. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction

=_A

instead of

=

.
Similarly, the "subset of" relation

subseteq

needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted

subseteq_A

. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation

in_A

which is a set.
Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple (X, Y, G), as normally a proper class can't be a member of an ordered tuple; or of course one can identify the function with its graph in this context.)
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context.

The number of binary relations

The number of distinct binary relations on an n-element set is 2^n² :
Notes:

The number of irreflexive relations is the same as that of reflexive relations

The number of (irreflexive transitive relations) is the same as that of partial orders

The number of strict weak orders is the same as that of total preorders

The total orders are the partial orders which are also total preorders. The number of preorders which are neither a partial order nor a total preorder is therefore the number of preorders minus the number of partial orders minus the number of total preorders plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

the number of equivalence relations is the number of partitions, which is the Bell number. The binary relations can be grouped into pairs (relation, ), except that for n = 0 the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement,, inverse complement).

Examples of common binary relations

Order relations, including strict orders:

Greater than
Greater than or equal to
Less than
Less than or equal to
Divides (evenly)
is a subset of

Equivalence relations:

Dependency relation, a symmetric, reflexive relation.

Independency relation, a symmetric, irreflexive relation.Further Information

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