Hey

i need help with a simple proof i'm stuck with

~ is a reflexiv relation on a set M

Show:

~ is a equivalence relation $\displaystyle $\Leftrightarrow$$ $\displaystyle $\forall x,y,z \in M \ (x \sim y) \wedge (x \sim z) \Rightarrow y \sim z$$

i did this:

"$\displaystyle $\Rightarrow$$" since ~ is an equivalence relation $\displaystyle $(x \sim y) \wedge (x \sim z) \Rightarrow (y \sim x) \wedge (x \sim z) \Rightarrow y \sim z$$

first argument because ~ is symmetric second because ~ is transitiv

"$\displaystyle $\Leftarrow$$"

Reflexivity: $\displaystyle $(x \sim y) \wedge (x \sim y) \Rightarrow (y \sim y)$$

Symmetry ?

Transitivity ?

are my thoughts correct? and how do i show the last two??

thx