Question

Prove or disprove: Every symmetric and transitive relation on a nonempty set is an equivalence relation.

Answer 1

I believe it should be disproved but I'm having difficulty figuring out a relation on a set A

Answer 2

So say A={a,b,c,d}.

Answer 3

find a relation that is symmetric and transitive, but not reflexive

Answer 4

R={(a,b),(b,a),(b,c),(a,c)}

Answer 5

that's easy it seems but I wonder if I am forgetting something.

Answer 6

i am not sure that one is symmetric

Answer 7

(c,a)

Answer 8

(c,b)

Answer 9

you have (b,c) but not (c, b(

Answer 10

it needs (c,b) too

Answer 11

I mean (c,a)

Answer 12

right?

Answer 13

lets try this

Answer 14

on the set \(\{a, b, c, \}\) define the relation \(\{(a,a), (a, b), (b, a), (b,b)\}\)
think you will see that it is symmetric and transitive, but not reflexive
it may seem like cheating, but it is not