Question

Total number of equivalence relations defined in the set $S=\{a,b,c\}$ is ?

Answer 1

Step 1.

• Consider the relation R1 = { (a,a) }
  

it is reflexive ,symmetric and transitive

• similarly R2= {(b,b)} , R3= {(c,c)} are reflexive ,symmetric and transitive
  

Step 2.

• Also R4 = { (a,a) ,(b,b),(c,c), (a,b),(b,a)}
  

it is reflexive as(x,x)∈R(a,a)∈R for all x∈a,b,ca∈1,2,3

it is symmetric as (x,y)∈R=>(y,x)∈R(a,b)∈R=>(b,a)∈R for all x,y∈a,b,ca∈1,2,3

also it is transitive as (a,b)∈R,(b,a)∈R=>(a,a)∈R(1,2)∈R,(2,1)∈R=>(1,1)∈R

Step. 3

• The relation defined by R = {(a,a), (b,b) , (c,c) , (a,b), (a,c),(b,a),(b,c), (c,a),(c,b)}
  

is reflexive symmetric and transitive

• Thus Maximum number of equivalance relation on set
  

A={a,b,c}A={1,2,3} is 5