Relations?

R is the set of all positive real numbers.  Define r @ x if and only if |r| = [x].  I don't have the correct symbols to use for it, but i do not mean absolute value.  |r| represents the greatest integer less than or equal to r.  [x] represents the smallest integer greater than or equal to x.


Is this relation an equivalence relation?  I'm struggling in why some relations ARE equivalence relations and some are not (some are partial order relations, for example).  Why is this or isn't it an equivalence relation?
liahowAsked:
Who is Participating?
I wear a lot of hats...

"The solutions and answers provided on Experts Exchange have been extremely helpful to me over the last few years. I wear a lot of hats - Developer, Database Administrator, Help Desk, etc., so I know a lot of things but not a lot about one thing. Experts Exchange gives me answers from people who do know a lot about one thing, in a easy to use platform." -Todd S.

Arthur_WoodCommented:
consider the following examples, and this should answer your question:

let r = 1.23 (or even 1.00000000000001)
let x = 0.23 (and      0.99999999999999)  for that matter

|r| = 1
[x] = 1

are r and x 'equivalent'???

AW
0
Harisha M GEngineerCommented:
'Equivalenence relation' has nothing to do with 'equivalence'...

A relation is said to be an equivalence relation if satisfies the following properties:

1) Reflexivity
2) Symmetry
3) Transitivity

http://en.wikipedia.org/wiki/Equivalence_relation

Reflexivity:
Let 0.5 be the number. Then, the greatest integer less than or equal to 0.5 is 0 and the smallest integer greater than or equal to 0.5 is 1
From this, you see that 0 <> 1 and hence, 0.5 cannot be a number in that set. So, your set should only consist of integers, and no real numbers.

And hence, reflexivity fails if the set contains real numbers. So, the given relation is not an equivalence relation under the set of reals.. however it is reflexive for set of integers and hence, the other two properties need to be checked..

Symmetry:
Now, consider 5 and 5 to be arbitrary elements in the set. 5 will be the greatest integer less than or equal to 5 and the smallest integer greater than or equal to 5. So they are same, and hence, it is symmetric.

Transitivity:
Let 5, 5 and 5 be the three elements, and 5 will be the result for all of them, and hence it is transitive too.

RESULT: The given relation is an equivalence relation under the set of integers
0

Experts Exchange Solution brought to you by

Your issues matter to us.

Facing a tech roadblock? Get the help and guidance you need from experienced professionals who care. Ask your question anytime, anywhere, with no hassle.

Start your 7-day free trial
NovaDenizenCommented:
The operation is define over the set of real numbers.  Symmetry fails for, say, 5.9 and 6.1.  

btw, the |x| and [y] functions you're referring to are usually referred to as floor(x) and ceil(y) when you don't have the notations available.
0
Cloud Class® Course: Python 3 Fundamentals

This course will teach participants about installing and configuring Python, syntax, importing, statements, types, strings, booleans, files, lists, tuples, comprehensions, functions, and classes.

Harisha M GEngineerCommented:
Symmetry fails for, say, 5.9 and 6.1.  

What about reflexivity ? It would also fail
0
NovaDenizenCommented:
I don't need to enumerate every reason that this relation is not an equivalence relation.  I just need to show one counterexample.
0
Harisha M GEngineerCommented:
Yeah, but generally, it is proved in order :)
0
liahowAuthor Commented:
Thanks for the tips!
0
It's more than this solution.Get answers and train to solve all your tech problems - anytime, anywhere.Try it for free Edge Out The Competitionfor your dream job with proven skills and certifications.Get started today Stand Outas the employee with proven skills.Start learning today for free Move Your Career Forwardwith certification training in the latest technologies.Start your trial today
Math / Science

From novice to tech pro — start learning today.

Question has a verified solution.

Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.

Have a better answer? Share it in a comment.