## Equivalence Relations

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
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 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.