Posted on 2006-04-06
Last Modified: 2008-03-17
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?
Question by:liahow
    LVL 44

    Expert Comment

    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'???

    LVL 37

    Accepted Solution

    '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

    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..

    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.

    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
    LVL 22

    Expert Comment

    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.
    LVL 37

    Expert Comment

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

    What about reflexivity ? It would also fail
    LVL 22

    Expert Comment

    I don't need to enumerate every reason that this relation is not an equivalence relation.  I just need to show one counterexample.
    LVL 37

    Expert Comment

    by:Harisha M G
    Yeah, but generally, it is proved in order :)

    Author Comment

    Thanks for the tips!

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