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# Mathematics | Introduction and types of Relations

Relation or Binary relation R from set A to B is a subset of AxB which can be defined as aRb â†” (a,b) â‚¬ R â†” R(a,b). A Binary relation R on a single set A is defined as a subset of AxA. For two distinct set, A and B with cardinalities m and n, the maximum cardinality of the relation R from A to B is mn.
Domain and Range: if there are two sets A and B and Relation from A to B is R(a,b), then domain is defined as the set { a | (a,b) â‚¬ R for some b in B} and Range is defined as the set {b | (a,b) â‚¬ R for some a in A}.

## Types of Relation:

1. Empty Relation: A relation R on a set A is called Empty if the set A is empty set.
2. Full Relation: A binary relation R on a set A and B is called full if AXB.
3. Reflexive Relation: A relation R on a set A is called reflexive if (a,a) â‚¬ R holds for every element a â‚¬ A .i.e. if set A = {a,b} then R = {(a,a), (b,b)} is reflexive relation.
4. Irreflexive relation : A relation R on a set A is called reflexive if no (a,a) â‚¬ R holds for every element a â‚¬ A.i.e. if set A = {a,b} then R = {(a,b), (b,a)} is irreflexive relation.
5. Symmetric Relation: A relation R on a set A is called symmetric if (b,a) â‚¬ R holds when (a,b) â‚¬ R.i.e. The relation R={(4,5),(5,4),(6,5),(5,6)} on set A={4,5,6} is symmetric.
6. AntiSymmetric Relation: A relation R on a set A is called antisymmetric if (a,b)â‚¬ R and (b,a) â‚¬ R then a = b is called antisymmetric.i.e. The relation R = {(a,b)â†’ R|a â‰¤ b} is anti-symmetric since a â‰¤ b and b â‰¤ a implies a = b.
7. Transitive Relation: A relation R on a set A is called transitive if (a,b) â‚¬ R and (b,c) â‚¬ R then (a,c) â‚¬ R for all a,b,c â‚¬ A.i.e. Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.
8. Equivalence Relation: A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive. i.e. relation R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)} on set A={1,2,3} is equivalence relation as it is reflexive, symmetric, and transitive. Number of equivalence relation in a set containing n elements is given by Bell number.
9. Asymmetric relation: Asymmetric relation is opposite of symmetric relation. A relation R on a set A is called asymmetric if no (b,a) â‚¬ R when (a,b) â‚¬ R.