Basics of relation | Types of relation | Reflexive | Irreflexive | Symmetric | AntiSymmetric | Asymmetric | Transitive | Equivalence

Relation: A binary relation from set A to set B is a subset of AxB (cartesian product of A and B).
Example:
A = {1, 2, 3, 4}
Find R = {(a, b) | a divides b} ?
Answer: R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}
Types of relation

Relation  Definition Example
Reflexive A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A A = {1, 2, 3}
R = {(1,1), (2,2), (3,3), (1,3), (1,4)}
Irreflexive A relation R on set A is called Irreflexive if no a∈A is related to a (aRa does not hold). A = {1, 2, 3}
R = {(1,3), (1,4)}
Symmetric A relation R on set A is called Symmetric if aRb implies bRa, ∀a∈Aand ∀b∈A. A = {1, 2, 3}
R = {(1,1), (2,2), (3,3), (1,3), (3,1)}
AntiSymmetric A relation R on set A is called Anti-Symmetric if aRb and bRa implies a=b, ∀a∈A and ∀b∈A. A = {1, 2, 3}
R = {(1,1), (2,2), (3,3), (1,3), (2,3}
Asymmetric A relation R on set A is called Asymmetric if for all a, b in A, if (a,b) is in R, then (b,a) is not in R. A = {1, 2, 3}
R = {(1,3), (2,3}
Transitive A relation R on set A is called Transitive if aRb and bRc implies aRc, ∀a,b,c ∈A A = {1, 2, 3}
R = {(1,2), (2,3), (1,3)}
Equivalence A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive. A = {1, 2, 3}
R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}
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