# equivalence relations

• Nov 23rd 2009, 05:43 PM
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?
• Nov 23rd 2009, 06:21 PM
Drexel28
Quote:

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.
• Nov 23rd 2009, 06:29 PM
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
• Nov 23rd 2009, 06:33 PM
Drexel28
Quote:

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 $\displaystyle nz=\bigcap_{x\in z}x$?
• Nov 23rd 2009, 06:36 PM
dont worrry about it and yes i did mean the intersection as u stated
• Nov 23rd 2009, 06:39 PM
Drexel28
Quote:

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?
• Nov 23rd 2009, 06:44 PM
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.
• Nov 23rd 2009, 06:48 PM
Drexel28
Quote:

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.
• Nov 23rd 2009, 06:51 PM
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.)