Basic quesion about equivalance classes

[pamparana]

Hello,

I am trying to brush up on some mathematics and realized that I have to start from the very bottom. I keep getting confused by the symbols and it has been an ardeous last few days!

Anyway, going through some stuff on set theory and came across the concept of equivalance classes. Let me see if I have understood this correctly:

If we have an equivalent relation on elements a and b of a set, than the set of all elements that are equivalent to a would be the equivalent class of element a. Does that sound right? I have a feeling I have understood it all wrong also because I am finding it hard to get used to the notations.

Then it goes on to say that if there is an equivalance relation on A and a, b are elements of set A, then either

[[a]] union [] = null set or [[a]] = []

This is not immediately obvious to me and I would be really grateful if someone can shed some light on this.

I would be grateful for any help you can give this old man :)

/Luca

 

[hokie1]

I think you mean intersection instead of union.

The equivalence classes do not have elements in common. If two elements of a set are equivalent, then a ~ b or a ~ c ~ d ... ~ b. Right? The element are related or there exists 1 or more applications of the transitive rule that show the elements are related. Related elements are equivalent elements.

The equivalence class of a is the same as the equivalence class of b if a~b or a ~ c ... ~ b. The equivalence class of a and the equivalence class of b have no elements in common if a and b are not equivalent.

 

[mXSCNT]

Best is to use an example: suppose we have a set A = {1,2,3,4,5} and a relation ~ such that a ~ b if and only if a mod 2 = b mod 2. Then we have:
1 ~ 3, 3 ~ 1
1 ~ 5, 5 ~ 1
3 ~ 5, 5 ~ 3

2 ~ 4, 4 ~ 2

You can see that [[3]] = [[5]] = [[1]] = {1,3,5} and [[2]] = [[4]] = {2,4}. Also, for example, [[2]] intersection [[3]] = null set.

My mental image is something like, I think of a set like a country (like the USA or Canada) and the equivalence classes like states or provinces within the country. The equivalence relation ~ is given by a ~ b if and only if a and b are both located in the same state; for example "Plymouth" ~ "Boston" because they are both located in Massachusetts, so they are in the same equivalence class. But "Los Angeles" and "New York City" are not related and are not in the same equivalence class, since they are not in the same state.

I hope this helps you understand it. Of course, in the end you have to use the basic axioms and definitions, but it helps to have a rough mental picture of what they are about.