Question

let is equal to (1,2,
3 )then number of equivalence relations containing( 1,2) is​

Answer 1

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 R  

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 R  

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 R  

2

​  

={(1,2),(2,1),(1,1),(2,2),(3,3),(2,3),(3,2),(1,3),(3,1)}

Hence, only two possible relation are there which are equivalence.