# The maximum number of equivalence relations on the set A={1,2,3} are

Toolbox:
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
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$(a,a) \in R$ for all $a \in {1,2,3}$
it is symmetric as $(a,b)\in R => (b,a) \in R$ for all  $a\in {1,2,3}$
also it is transitive as $(1,2)\in R , (2,1)\in R => (1,1)\in R$
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

edited Mar 27, 2013 by meena.p