Unnecessary property in definition of equivalence relation

Question

Possible Duplicates:
Symmetric, Transitive and reflexive

Why isn't reflexivity redundant in the definition of equivalence relation?

Dependence of Axioms of Equivalence Relation?

Let $X$ a set and let $\sim$ a binary relation in $X$. $\sim$ is called a equivalence relation if:

1. $\forall x\in X$ we have $x\sim x$.
2. $\forall x,y\in X$ if $x\sim y$ then $y\sim x$.
3. $\forall x,y,z\in X$ if $x\sim y$ and if $y\sim z$ then $x\sim z$.

I think that 1 is unnecessary because by 2 we have that $x\sim y \Leftrightarrow y\sim x$. Then by 3. we have that $x\sim y$ and $y\sim x$ then $x\sim x$. Then 2,3 $\Rightarrow$ 1.

Am I right?

Answer 1

You are right unless there is some $x$ that is unrelated to the other elements. If $x\sim y$ is false for all $y$, then 2 and 3 might both hold, but 1 does not.

In particular, the empty relation, which has $x\not\sim y$ for all $x$ and $y$, is symmetric and transitive, but not reflexive.

Answer 2

no

what if there is no such $y$?