Any help anyone? thank you
Hey guys, question... Let A = {1,2,3,4,5,6,7,8,9,10}. Define a relation R on A by writing (x,y) R iff 3|(x-y).
a) show that R is an equivalence relation on A.
My answer:
Is it reflexive? Yes! since x-x = 0 and 0 is divisible by 3. (Eg x = 6)
Is it symmetric? Yes! since if x-y is divisible by 3, then y-x is divisible by 3. (Eg, x=8, y=2)
Is it transitive? Yes! since if 3|(x-y) and 3|(y-z), then 3|(x-z). (Eg. x=10, y=1 and z=4).
Is this correct? thanks..
You're correct. But leave out the examples. They're irrelevant to the proof, in the sense that just because something holds for example x,y,z in A doesn't entail that it holds for all x,y,z in A.
Here's how I would write it:
x-x = 0, and 3*0 = 0, so 3|x-x. So we have reflexivity.
Suppose 3|x-y. So let 3*z = x-y. So 3*-z = y-x. So 3|y-x. So we have symmetry.
Suppose 3|x-y and 3|y-z. So let 3*v = x-y and 3*w = y-z.
But x-z = (x-y)+(y-z) = (3*v)+(3*w) = 3(v+w). So 3|x-z. So we have transitivity.
I don't know what you mean.
It is possible to partition A into subsets by equivalence class, as follows
{1,4,7,10}
{2,5,8}
{3,6,9}
The same can be done for the set of integers.
Or you can talk about the set of all equivalence classes, which is often expressed in terms of common residues, like this
or this
{0,1,2}
Is that what you had in mind?