1. ## equivalence relations

say we have a set W. We also have a non-empty set Z such that each element of Z is an equivalence relation on W.

Must show that the nZ is an ER in W. (where n is intersection)
how would this be proven, like i know that u must prove that it is reflexive, transitive and symmetric but how exactly would u start it?

2. Originally Posted by g rad23
say we have a set W. We also have a non-empty set Z such that each element of Z is an equivalence relation on W.

Must show that the nZ is an ER in W. (where n is intersection)
how would this be proven, like i know that u must prove that it is reflexive, transitive and symmetric but how exactly would u start it?
I am sorry, I don't understand what you mean.

3. my question is
let W be a set.
Let Z be a non empty set such that each element of Z is an equivalence relation on W

i must show that the nZ is an equivalence relation on W. (where nZ is the intersection of Z)

how would i go about proving this is true?

i understand that i have to prove that the relation is reflexive, anti-symmetric and transitive, but how would i do that

4. Originally Posted by g rad23
my question is
let W be a set.
Let Z be a non empty set such that each element of Z is an equivalence relation on W

i must show that the nZ is an equivalence relation on W. (where nZ is the intersection of Z)

how would i go about proving this is true?

i understand that i have to prove that the relation is reflexive, anti-symmetric and transitive, but how would i do that
I'm sorry. I don't mean to sound ignortant, but I was unsure of notation. Do you mean $\displaystyle nz=\bigcap_{x\in z}x$?

5. dont worrry about it and yes i did mean the intersection as u stated

6. Originally Posted by g rad23
dont worrry about it and yes i did mean the intersection as u stated
Maybe some other member can help you better than I, but I don't really undertsand what this question could possibly mean? What is the intersection of two equivalence relations? Unless, you mean the intersection of all the partitions induced by the relations?

7. well im not sure myself this is the question i was given for my hmwk. well im assuming its intersection it looks exactly like the symbol for intersection but has no limit.

8. Originally Posted by g rad23
well im not sure myself this is the question i was given for my hmwk. well im assuming its intersection it looks exactly like the symbol for intersection but has no limit.
Ok, now that I think about it I believe that I understand what this is saying. if $\displaystyle \sim$ is an equivalence relation on $\displaystyle E$ then $\displaystyle \sim$ is charcterized by $\displaystyle R=\left\{(a,b)\in E\times E:a\sim b\right\}$. Maybe that if we consider the different relations on $\displaystyle E$ to be charchertized by $\displaystyle R_1,R_2,\cdots$ (not necessarily countable..I just wrote it that way for clarity) then perhaps they mean to show that $\displaystyle R_1\cap R_2\cdots=R$ is an the characterization of an equivalence relation. Does that sound right? Another member may swoop in an answer this btw.

9. umm it might be right im not sure i thought in order to prove it was an equivalence relation on something one would have to prove that the relation is reflexive, transitive and symmetric

10. The gist of the problem is to prove the following. Let $\displaystyle W$ be a set and let $\displaystyle R_1$ and $\displaystyle R_2$ be relations (i.e., subsets of $\displaystyle W\times W$) on $\displaystyle W$. Moreover, suppose that $\displaystyle R_1$ and $\displaystyle R_2$ are equivalence relations. Show that $\displaystyle R_1\cap R_2$ is an equivalence relation a well.

(This solves the problem when the set $\displaystyle Z$ of equivalence relations has size 2, or, in fact, any finite size. To be strict, one has to prove this also for infinite $\displaystyle Z$, but the proof is essentially the same.)

This should be easy to show by definition.

11. Yeah, i agree with emakarov!