Math Help - Equivalence classes

1. Equivalence classes

I am learning about equivalence classes and I need little help beacuse it's a little bit abstract for me to understand it.

This is how I understand it, and don't mind if it's not strict math.

Equivalence classes are forming subsets of equal elements from set. Since there are elements in set that are in equivalence relation it is possible to group those equal elements in groups which are actually called equivalence classes. For example, relation "=" form equivalence classes [1,1],[2,2],[3,3],[4,4],[5,5]... in N. In books there is often an example of relation congruence modulo 2 for explaining equivalence classes. So, equivalence classes are actually a way to group all elements that are equal among themselves.

Am I right, did I understand equivalence classes correctly?

Now, I need also explanations of definition in book about equivalence class.

Definiton of equivalence class:
If $\sim$ is relation of equivalence, then class of element $x$, shown as symbol $C_x$, defines like: $C_x = \{ y|x \sim y\}$

2. I think you're mixing up the relation and the classes.

Any relation can be expressed as a set of ordered pairs, where (a,b) is in R if and only if the relation aRb is true. So equality is expressed as the set (1,1),(2,2),(3,3) ... Similarly the relation < can be expressed as the set (1,2),(1,3),(1,4),...(2,3),(2,4),...(3,4),... .

If a relation is an equivalence (RST: reflexive, symmetric, transitive) then it can also be defined by the classes of things which are equivalent. For equality this is the collection of singletons {1}, {2}, {3}, ...

Congruence modulo 2 is also known as parity: a==b iff their difference a-b is even. The relation is expressed as (1,1),(1,3),(1,5),...,(3,3,),(3,5),...,(2,2),(2,4) ,(2,6),...,(4,4),(4,6),... and the two classes are {1,3,5,...} and {2,4,6,...}.

3. Originally Posted by rgep
I think you're mixing up the relation and the classes.

Any relation can be expressed as a set of ordered pairs, where (a,b) is in R if and only if the relation aRb is true. So equality is expressed as the set (1,1),(2,2),(3,3) ... Similarly the relation < can be expressed as the set (1,2),(1,3),(1,4),...(2,3),(2,4),...(3,4),... .

If a relation is an equivalence (RST: reflexive, symmetric, transitive) then it can also be defined by the classes of things which are equivalent. For equality this is the collection of singletons {1}, {2}, {3}, ...

Congruence modulo 2 is also known as parity: a==b iff their difference a-b is even. The relation is expressed as (1,1),(1,3),(1,5),...,(3,3,),(3,5),...,(2,2),(2,4) ,(2,6),...,(4,4),(4,6),... and the two classes are {1,3,5,...} and {2,4,6,...}.
Maybe I didn't explain good, but that is actually more-less what I wanted to say.