Question

let A={1,2,3}. find the number of equivalence relations containing (1,2) ?

Answer 1

Since equivalence relations must be symmetric, if (1,2) is in the relation then so is (2,1). It must also contain (1,1), (2,2), and (3,3), because the relation must be reflexive. So we have one relation: \(\{(1,1),(2,2),(3,3),(1,2),(2,1)\}\), and another relation \(\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3)(3,2)\}\). So, two relations.

Answer 2

only one

Answer 3

(1,2) comes only once.

Answer 4

is it (1,2) ?

Answer 5

@shruti it's correct

Answer 6

have you asked for subset (1,2) or combinations of 1and 2.

Answer 7

@shruti Have you understood my answer? There are two equivalence relations that contain (1,2). One of them is the one which partitions the set into {1,2} and {3}, the other is the relation in which all three elements are equivalent.

Answer 8

yeah . i got it.thanks @nbouscal

Answer 9

so the number of ways to partition the set is also the number of ways to make equivalence relations

Answer 10

Yes, equivalence relations and partitions are the exact same thing, just two different ways of looking at it.

Answer 11

Also {1,3} {2}

Answer 12

{1} {2} {3} <-> { (1,1) (2,2) (3,3) }
{1,2} {3} <-> { (1,1) (1,2) (2,1) ) (2,2) (3,3) }
{1} {2,3} <-> { (1,1) (2,2) (2,3) (3,2) (3,3) }
{1,3} {2} <-> { (1,1) (2,2) (1,3) (3,1) (3,3) }
{1,2,3} <-> { (1,1)(2,2) (3,3) (1,2) (2,1) (1,3)(3,1) (2,3) (3,2)}

Answer 13

and just as there are 2 partitions that contain {1,2} as a subset, there are two equivalence relations that contain (1,2) as a member

Answer 14

hey teach me more, this is cool stuff. how do we count the number of partitions of a set,

Answer 15

yeah..it is.just a view to understand is needed.