Here you will learn what is inverse relation, identity relation and posets in relation with example.
Letโs begin โ
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 \(R^{-1}\)
andย Range of R = Domain of \(R^{-1}\)
\(\therefore\)ย ย \(R^{-1}\) = {(b, a) | (a, b) \(\in\) R}
A relation R is defined on the set of 1st ten natural numbers.
e.g.ย ย 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.ย 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), โฆโฆ}
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.
