For the set of cities on a map, consider the relation x r y if and only if city x is connected by a road to city y. A city is considered to be connected to itself, and two cities are connected even though there are cities on the road between them. Is this an equivalence relation or a partial ordering? Explain.

# Equivalene relation

Let R be a relation on set A . If R is reflexive, symmetric, and transitive then it is said to be equivalence relation. Consequently, two elements a and b related by an equivalence relation are said to be equivalent .

Example - Show that relation R ={(a,b)|a=b(mod m)} is an equivalence relation.

Solution -

- Reflexive - for element a-a=0 is divisible by m.

a=a(mod m) . So congruence modulo m is reflexive. - Symmetric - For any two elements a and b , if (a,b) €R i.e. Congruence Modulo m is Symmetric.
- Transitive - For any three elements a,b and c if (a,b), (b,c) € R then

(a-b) mod m =0

(b-c) mod m =0

Adding both above equations (a-c)mod m =0

So R is transitive

Therefore R is an equivalence relation