Equivalence Classes Homework Help - #1 & #5

[cgjolberg]

Homework Statement

If you follow this link
http://www.math.tamu.edu/~ciken/teaching/spring2014/math302/practice%20midterm%202.pdf
There are several optional problems that have been posted for studying for my exam. I figured it would be easier to read the original than have me try to retype it.
The only ones I am concerned with are #1 and #5

Homework Equations

The Attempt at a Solution

For number one I got the first part just fine.
Would it be 4 equivalence classes? Not sure if I solved it correctly.
Also, how would I go about proving this fact if I got it right.
While of course a proof would be welcome, if you have a link to a resource that would show me how or just any tips really would be great!
For number 5.
Would it be 10 equivalence classes?

Post anything you know. Unfortunately time is of the essence because exam is soon.
Wish this had been posted before:(

Thanks for the help!

 

[abitslow]

Your attempt at a solution is a guess? ("would it be 4..."). I guess so.

 

[kduna]

You can in fact make #5 more general.

x ~ y iff x = y + n*k for some k.

There are n equivalence classes.

 

[jbunniii]

For problem 1, let $|E|$ denote the number of edges. Surely there is at least one connected graph for each of $|E| = 3,4,5,6$. Your solution would imply that there is exactly one equivalence class for each of these values of $|E|$. Look more carefully at $|E| = 4$...