Question

Find maximum number of equivalence relations on the set A = {1, 2, 3}.

Answer 1

Answer:

A relation R in a set A is called reflexive, if (a,a)∈R for every a∈A

A relation R in a set A is called symmetric, if (a1,a2)∈R⇒(a2,a1)∈Rfora1,a2∈A

A relation R in a set A is called transitive, if (a1,a2)∈R and (a2,a3)∈R⇒(a1,a3)∈R for alla1,a2,a3∈A

step 1.

consider the relation R 1 = { (1,1) }

it is reflexive ,symmetric and transitive

similarlyR 2= {(2,2)} , R  3= {(3,3)} are reflexive ,symmetric and transitive

Step 2.

Also R 4 = {  (1,1) ,(2,2),(3,3), (1,2),(2,1)}

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

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

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

Step. 3

The relation defined by R = {(1,1), (2,2) , (3,3) , (1,2), (1,3),(2,1),(2,3) (3,1),(33,2)}

is reflexive symmetric and transitive

 

Thus Maximum number of equivalance relation on set A={1,2,3} is 5

Answer 2

Answer:

Step-by-step explanation:we know that equivalence means the relation is symmetric,transative,reflexive.

Let's take

R1={(1,1),(2,2),(3,3)}

R2={(1,1),(2,2),(3,3),(1,2),(2,1)}

R3={(1,1),(2,2),(3,3),(2,3),3,2)}

R4={(1,1),(2,2),(3,3),(1,3),(3,1)}

R5={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)}

So,the maximum equivalence relations on the set A is 5