1. ## 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.

2. For #1.
Reflexive: is it true that x-x=0 for all x? If so, the relation is reflexive.
Symmetric: is it true that if x-y=0 then y-x=0? If so, the relation is Symmetric.
Transitive: is it true that if x-y=0 & y-z=0 then x=z=0? If so the relation is transitive.

Do note that x-y=0 if and only if x=y.

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

4. And number 2 is reflexive and transitive?

5. No. #1 are all three.
And
#2 only transitive.

$\displaystyle x-x=0$ thus it is reflexsive for all $\displaystyle x\in S$.

If $\displaystyle x-y=0$ then $\displaystyle y-x=0$, thus it is symterric.

If $\displaystyle x-y=0$ and $\displaystyle y-z=0$ then $\displaystyle x-y+y-z=x-z=0$, 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 $\displaystyle x<z$

6. Thanks for the explanation.