Functions question

[royal lodge]

let a= 1 2 3 . then find the number of equivalence relations containing (1 2) is?

Im getting an answer =3, the book says its 2 and websites i checked say its 1

[marsh hemlock]

first relation is (1,1),(2,2),(3,3),(1,2),(2,1)

second is (1,1),(2,2),(3,3),(1,3),(3,1),(3,2),(2,3),(1,2),(2,1)

i have to send the answer for me 2 explaint his

so basicaly first relation satisfys alll 3 critera

and because we dont have a relation of 1,3 in first we dont need to add the other 4

rest transitive symetric its clerly visible

for the second relation

we add 3,2 2,3 for symetric

and 1,3 3,1 for transitive

because 1 has relation iwth 2

2 has relation with 3

so 1 has also relation with 3

i beleive its from ncert