Question

Let A ={1,2,3}.Then, number of equivalence relations containing (1,2) is
(a)1
(b)2
(c)3
(d)4

Answer 1

It is given that A ={1,2,3}.
An equivalence relation is reflexive, symmetric and transitive.
The smallest equivalence relation containing (1,2) is given by,
$R_1=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\}$
Now, we are left with only four pairs i.e., (2,3),(3,2),(1,3), and (3,1).
If we add any one pair [say(2,3)] to $R_1$, then for symmetry we must add (3,2).
Also, for transitivity we are required to add (1,3)and (3,1).
Hence, the only equivalence relation (bigger than $R_1)$ is the universal relation.
This shows that the total number of equivalence relations containing (1,2) is two.
Thus, the correct answer is (b).