# equivalence relations

• Dec 22nd 2006, 07:27 AM
papa_chango123
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.
• Dec 22nd 2006, 08:19 AM
Plato
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.
• Dec 22nd 2006, 09:12 AM
papa_chango123
So would I be safe to assume that #1 is only reflexive?
• Dec 22nd 2006, 09:13 AM
papa_chango123
And number 2 is reflexive and transitive?
• Dec 22nd 2006, 10:07 AM
ThePerfectHacker
No. #1 are all three.
And
#2 only transitive.

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

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

If $x-y=0$ and $y-z=0$ then $x-y+y-z=x-z=0$, thus it is transitive.

Similary with <
It is not true that,
$x for all $x\in S$

It is not true that,
If $x then $y

It is true that,
$x and $y then $x
• Dec 22nd 2006, 01:21 PM
papa_chango123
Thanks for the explanation.