Is a Symmetric and Transitive Relation Always Reflexive?

[jeff1evesque]

Statement:
Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive.


Ideas:
If our relation ~ is transitive, then we know: a~b, and b~a \Rightarrow a~a.
Therefore our relation ~ is reflexive, since b~c and c~b \Rightarrow b~b, and c~a and a~c \Rightarrow c~c.


Proof:
Can the above (idea) constitute a proof in itself?

Thanks,

Jeffrey

 

[jeff1evesque]

Actually I thought about it a little, and came up with a proof. But can someone critique it and let me know if it's actually alright.


Proof
We know ~ is symmetric.
Therefore, \exists a,b,c \in A such that

if a~b, then b~a,
and if b~c, then c~b,
and if c~a, then a~c.​

But we also know our relation ~ is transitive.
Therefore,

if a~b, and b~a, then a~a, (#1)
and if b~c, and c~b, then b~b, (#2)
and if c~a, and a~c, then c~c. (#3)​

By (#1), (#2), and (#3) we know our given relation is reflexive.