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Properties of Binary Relation in a Set

In this tutorial, we will learn about the relation, and properties of binary relation in a set. By Prerana Jain Last updated : May 09, 2023

Relation

The word relation suggests some familiar example relations such as the relation of father to son, mother to son, brother to sister etc. Familiar examples in arithmetic are relation such as "greater than", "less than", or that of equality between the two real numbers.

Here, we shall only consider relation called binary relation, between the pairs of objects. Before we give a set-theoretic definition of a relation we note that a relation between two objects can be defined by listing the two objects an ordered pair.

Definition

Any set of ordered pairs defines a binary relations. We shall call a binary relation simply a relation. It is sometimes convenient to express the fact that particular ordered pair say (x,y) E R where, R is a relation by writing xRY which may be read as "x is a relation R to y".

Example

The relation of father to his child can be described by a set , say ordered pairs in which the first member is the name of the father and second the name of his child that is:

F = { (x , y) |x is the father of y}

Domain

Let, S be a binary relation. The set D(S) of all objects x such that for some y, (x,y) E S is said to be the domain of S.

Range

The set R(S) of all objects y such that for some x, (x,y) E S said to be the range of S.

Let r A B be a relation

    DOM(R) = {a|(a, b)E R for some b E B} 
    Range(R) = {b |(a, b) E R } for some

Properties of Binary Relation in a Set

There are some properties of the binary relation:

1. A binary relation R is in set X is reflexive if , for every x E X , xRx, that is (x, x) E R or R is reflexive in X <==> (x) (x E X -> xRX).
   The relation =< is reflexive in the set of real number since for nay x we have x<= X similarly the relation of inclusion is reflexive in the family of all subsets of a universal set.
2. A relation R is in a set X is symmetric if for every x and y in x whenever xRy then yRX that is R is a symmetric in x.
   The relation <= and < are not symmetric i the set of real number while the relation of equality is.
3. A relation R in a set x is transitive if for every x, y and z in X whenever xRy and yRx then xRz that is R is transitive in X.
   The relation <= < and = are transitive in the set of real numbers. The relations and equality are also transitive in the family of a subset of a universal set.

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