Question

How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all? Justify your answer.

Answer 1

Hint: A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation.

Complete step-by-step answer:
Total possible pairs = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}
Reflexive means (a,a) should be in relation.
So, (1,1), (2,2), (3,3) should be in relation.
Symmetric means if (a,b) is in relation, then (b,a) should be in relation.
So, since (1,2) is in relation, (2,1) should also be in relation.

Transitive means if (a,b) is in relation and (b,c) is in relation, then (a,c) is in relation.
So, if (1,2) is in relation and (2,1) is in relation, then (1,1) should be in relation.

Relation ${{\text{R}}_{\text{1}}}$
 = {(1,2), (2,1), (1,1), (2,2), (3,3)}
Total possible pairs = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}
So, smallest relation is ${{\text{R}}_{\text{1}}}$
 = {(1,2), (2,1), (1,1), (2,2), (3,3)}
If we add (2,3)
then we have to add (3,2) also, as it is symmetric
but, as (1,2) & (3,2) are there, we need to add (1,3) also, as it is transitive

As we are adding (1,3) we should add (3,1) also, as it is symmetric
Relation ${{\text{R}}_2}$
 = {(1,2), (2,1), (1,1), (2,2), (3,3), (2,3), (3,2), (1,3), (3,1)}
Hence, only two possible relations are there which are equivalence.


Note- The concept of relation is used in relating two objects or quantities with each other. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. and also, a relation in set A is a subset of A × A. Thus, A × A is two extreme relations.