
equivalence relations
I am having a very hard time understanding how to tell whether relations are
reflexive, symmetric, and transitive.
reflexive x = y
symmetric x = y and y = x
transitive x = y and y = z
I just dont know how to go about working the problems. Here are the problems.
Determine which of the reflexive, symmetric, and transitive properties are satisified by the given relation R defined on se S, and stat whether R is an equivalence relation on S.
S = {1,2,3,4,5,6,7,8} and x R y means that x  y = 0.
S = {1,2,3,4,5,6,7,8} and x R y means that x > y
If someone can help me work thru the first one I can probably figure out the second one.

For #1.
Reflexive: is it true that xx=0 for all x? If so, the relation is reflexive.
Symmetric: is it true that if xy=0 then yx=0? If so, the relation is Symmetric.
Transitive: is it true that if xy=0 & yz=0 then x=z=0? If so the relation is transitive.
Do note that xy=0 if and only if x=y.

So would I be safe to assume that #1 is only reflexive?

And number 2 is reflexive and transitive?

No. #1 are all three.
And
#2 only transitive.
$\displaystyle xx=0$ thus it is reflexsive for all $\displaystyle x\in S$.
If $\displaystyle xy=0$ then $\displaystyle yx=0$, thus it is symterric.
If $\displaystyle xy=0$ and $\displaystyle yz=0$ then $\displaystyle xy+yz=xz=0$, thus it is transitive.
Similary with <
It is not true that,
$\displaystyle x<x$ for all $\displaystyle x\in S$
It is not true that,
If $\displaystyle x<y$ then $\displaystyle y<x$
It is true that,
$\displaystyle x<y$ and $\displaystyle y<z$ then $\displaystyle x<z$

Thanks for the explanation.