What is Inverse Relation – Definition and Example

Here you will learn what is inverse relation, identity relation and posets in relation with example.

Let’s begin –

Definition of Inverse Relation

If relation R is defined from A to B, then inverse relation would be defined form B to A, i.e.

R : A $\rightarrow$ B $\implies$ aRb where a $\in$ A, b $\in$ B

$R^{-1}$ : B $\rightarrow$ A $\implies$ bRa where a $\in$ A, b $\in$ B

Domain of R = Range of $R^{-1}$

and  Range of R = Domain of $R^{-1}$

$\therefore$   $R^{-1}$ = {(b, a) | (a, b) $\in$ R}

A relation R is defined on the set of 1st ten natural numbers.

e.g.   N is a set of first 10 natural numbers.

aRb $\implies$ a + 2b = 10

R = {(2, 4), (4, 3), (6, 2), (8, 1)}

$R^{-1}$ = {(4, 2), (3, 4), (2, 6), (1, 8)}

Identity Relation

A relation defined on a set A is said to be an identity relation if each and every element of A is related to itself & only to itself.

e.g.  A relation defined on the set of natural numbers is

aRb $\implies$ a = b where a & b $\in$ N

R = {(1, 1), (2, 2), (3, 3), ……}

R is an identity relation

Posets

A relation R on a set P is called an partial relation order if it is reflexive, antisymmetric and transitive. That means that for all x, y and z in P we have:

• x R x;
• if x R y and y R x, then x = y;
• if x R y and y R z, then x R z.

The pair (P, R) is called a partially ordered set, or for short, a poset.

Two elements x and y in a poset (P, R) are called comparable if x R y or y R x.

If any two elements x,y $\in$ P are comparable, so we have x R y or y R x, then the relation is called a linear order.