Prove by examples that every function can be binary relation but it is not necessary that ever binary relation can be function
Answers
Answer:
In mathematics, 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 information of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set. 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]
An example of a binary relation is the "divides" relation over the set of prime numbers P and the set of integers Z, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.
Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
the "is greater than", "is equal to", and "divides" relations in arithmetic;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.
A function may be defined as a special kind of binary relation.[3] Binary relations are also heavily used in computer science.
A binary relation over sets X and Y is an element of the power set of X × Y. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X × Y.
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder,[4] Clarence Lewis,[5] and Gunther Schmidt.[6] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.
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.
The terms correspondence,[7] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.
Step-by-step explanation:
Given sets X and Y, the Cartesian product X × Y is defined as {(x, y) | x in X and y in Y}, and its elements are called ordered pairs.
A binary relation R over sets X and Y is a subset of X × Y.[1][8] The set X is called the domain[1] or set of departure of R, and the set Y the codomain or set of destination of R. In order to specify the choices of the sets X and Y, some authors define a binary relation or correspondence as an ordered triple (X, Y, G), where G is a subset of X × Y called the graph of the binary relation. The statement (x, y) in R reads "x is R-related to y" and is denoted by xRy.[4][5][6][note 1] The domain of definition or active domain[1] of R is the set of all x such that xRy for at least one y. The codomain of definition, active codomain,[1] image or range of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain of definition and its codomain of definition.[10][11][12]
When X = Y, a binary relation is called a homogeneous relation (or endorelation). To emphasize the fact that X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.[13][14][15]
In a binary relation, the order of the elements is important; if x ≠ y then xRy, but yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.
Example Edit
2nd example relation
ball car doll cup
John + − − −
Mary − − + −
Venus − + − −
1st example relation
ball car doll cup
John + − − −
Mary − − + −
Ian − − − −
Venus − + − −
The following example shows that the choice of codomain is important. Suppose there are four objects A = {ball, car, doll, cup} and four people B = {John, Mary, Ian, Venus}. A possible relation on A and B is the relation "is owned by", given by R = {(ball, John), (doll, Mary), (car, Venus)}. That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of A × {John, Mary, Venus}, i.e. a relation over A and {John, Mary, Venus}.