How many classes does the equivalence relation partition the set?

Question

Looking for help on B) How many classes does the equivalence relation partition set X

Consider the relations $R$ and $S$, defined on the set $X = \{1, 2, . . . , 99\}$ as follows.

$xRy \iff x + y$ is a multiple of $11$,

$xSy \iff x − y$ is a multiple of $11$.

A) One of $R$ and $S$ is an equivalence relation, the other is not. Determine which is which and justify your answers.

B) Into how many classes does the equivalence relation partition set $X$?

So far I've determined that S is the equivalence relation as R isn't reflexive or transitive. My only attempt at B is that there is 11-1=10 non-zero congruence classes, so does each one correspond to a non-zero equivalence class?

Any help is appreciated!

Answer 1

Just count them.

$1 R 12 R 23 R 34 R .... R 90$

$2 R 13 R 24 R 35 R .... R 91$

....

$10 R 21 R 32 R 43 R..... R 98$

$11 R 22 R 33 R 44 R ..... R 99$

That's 11.

It doesn't make any sense to subtract the "zero" equivalence class for 2 reasons.

1) The "zero" equivalence class is STILL an equivalence class so why they heck would you omit it? NOWHERE in the question does it say ANYTHING about how many non-zero equivalence classes; it ask how many equivalence classes. And $\{11,22,33,44,55,66,77,88,99\}$ is certainly an equivalence class, isn't it?

2) Since no method of "addition" has been defined or discussed, there is no meaning to defining any one of the equivalence classes as be a "zero" equivalence class. IF we were to define $\{x|x R c\} + \{x|x R d\} = \{x| x R (d+c\pm 11k \text{ for some integer } k)\}$ and define $[0]$ as the equivelence clas so that $[0] + \{x|x R c\} = \{x|x R c\}$ then, yes, $\{11,22,33,44,55,66,77,88,99\} = [0]$. But again, so what, it's still an equivalence class, isn't it?

Answer 2

To show $S$ is an equivalence relation you can either verify directly that it is reflexive, transitive, and symmetric or demonstrate that the relation partitions the set and use the fundamental theorem of equivalence relations. If you can demonstrate the existence of the congruence classes you have such a partition and that's sufficient to demonstrate that you have an equivalence relation. As you've deduced there are eleven such classes, and ten of them do not contain zero. I would also encourage you to consider the first method of directly verifying the three defining properties as it's both instructive and relatively easy.