Symmetric, Transitive and reflexive
Let $X$ a set and let $\sim$ a binary relation in $X$. $\sim$ is called a equivalence relation if:
- $\forall x\in X$ we have $x\sim x$.
- $\forall x,y\in X$ if $x\sim y$ then $y\sim x$.
- $\forall x,y,z\in X$ 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?