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Math Help - relations help (2)

  1. #1
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    relations help (2)

    hey guys, question:

    Determine whether the following relation is a reflexive, symmetric or transitive. If any is an equivalence relation, describe its equivalence classes.

    xRy iff x has the same integer part as y, on <----- I need help on what this relation is asking? thanks
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    Quote Originally Posted by jvignacio View Post
    hey guys, question:

    Determine whether the following relation is a reflexive, symmetric or transitive. If any is an equivalence relation, describe its equivalence classes.

    xRy iff x has the same integer part as y, on <----- I need help on what this relation is asking? thanks
    The integer part of x is the integer n such that n\le x < n+1. (There is exactly one such integer for each real x.) It is usually denoted by [x].

    For example:
    [3.14]=3
    [0.52]=0
    [1]=1
    [-1.26]=-2

    So 2 and 2.5 are in relation, since [2]=2=[2.5], but 1.5 and 2.5 are not.

    Hope this helps.
    Last edited by Plato; April 18th 2010 at 06:14 AM. Reason: LaTeX fix
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  3. #3
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    Quote Originally Posted by kompik View Post
    the integer part of x is the integer n such that n\le x < n+1. (there is exactly one such integer for each real x.) it is usually denoted by [x].

    For example:
    [3.14]=3
    [0.52]=0
    [1]=1
    [-1.26]=-2

    so 2 and 2.5 are in relation, since [2]=2=[2.5], but 1.5 and 2.5 are not.

    Hope this helps.
    Is that like saying the next full integer down?
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    Quote Originally Posted by jvignacio View Post
    Is that like saying the next full integer down?
    Exactly.
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  5. #5
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    Quote Originally Posted by kompik View Post
    Exactly.
    So in this case,

    - It IS reflexive since x has the same integer part as x.
    - It IS symmetric since if x has the same integer part as y then y has the same integer part as x.
    - It IS transitive since if x has the same integer part as y and if y has the same integer part as z then x has the same integer part as z.

    correct?
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  6. #6
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    Quote Originally Posted by jvignacio View Post
    So in this case,

    - It IS reflexive since x has the same integer part as x.
    - It IS symmetric since if x has the same integer part as y then y has the same integer part as x.
    - It IS transitive since if x has the same integer part as y and if y has the same integer part as z then x has the same integer part as z.

    correct?
    Yes, it is an equivalence relation.

    Another way to see this is to notice that you're in fact given a decomposition of R and every decomposition gives you an equivalence relation. (But if you haven't heard much about the correspondence between equivalences and decompositions at your lessons, you should perhaps ignore this comment.)
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  7. #7
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    Quote Originally Posted by kompik View Post
    Yes, it is an equivalence relation.

    Another way to see this is to notice that you're in fact given a decomposition of R and every decomposition gives you an equivalence relation. (But if you haven't heard much about the correspondence between equivalences and decompositions at your lessons, you should perhaps ignore this comment.)
    Yeah ive never herd of that... Not yet anyway. What would the equivalence class be in this case now that its an equivalence relation? All numbers?
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  8. #8
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    Quote Originally Posted by jvignacio View Post
    Yeah ive never herd of that... Not yet anyway. What would the equivalence class be in this case now that its an equivalence relation? All numbers?
    No.
    Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
    This is equivalent to
    [x]=[3.14]=3.
    And what are the numbers such that [x]=3? Precisely the numbers from the interval \langle 3,4), right?
    Can you find the remaining classes?
    Last edited by kompik; April 18th 2010 at 05:48 AM.
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  9. #9
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    Quote Originally Posted by kompik View Post
    No.
    Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
    This is equivalent to
    [x]=[3.14]=3.
    And what are the numbers such that [x]=3? Precisely the numbers from the interval \lange 3,4), right?
    Can you find the remaining classes?
    Sorry whats in the interval for [x]=3? theres a latex error..
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  10. #10
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    Quote Originally Posted by jvignacio View Post
    Sorry whats in the interval for [x]=3? theres a latex error..
    Sorry, did not notice that. I've edited my post.
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  11. #11
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    Quote Originally Posted by kompik View Post
    No.
    Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
    This is equivalent to
    [x]=[3.14]=3.
    And what are the numbers such that [x]=3? Precisely the numbers from the interval \langle 3,4), right?
    Can you find the remaining classes?
    Ok I understand the equivalent class of 3.14 and the numbers such that [x]=3 are all numbers between 3 and 4 but Which remaining classes are you referring too? Since my question is a x and y question, how can I write this in a general form. If you get what I mean...
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    Quote Originally Posted by jvignacio View Post
    Ok I understand the equivalent class of 3.14 and the numbers such that [x]=3 are all numbers between 3 and 4 but Which remaining classes are you referring too? Since my question is a x and y question, how can I write this in a general form. If you get what I mean...
    If I were your teacher, I would expect answer to be something like:
    The equivalent classes are intervals .... (fill in the dots) for n\in\mathbb{N}.
    (I guess the example with 3.14 might help you to see what to fill in.)
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  13. #13
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    Quote Originally Posted by kompik View Post
    If I were your teacher, I would expect answer to be something like:
    The equivalent classes are intervals .... (fill in the dots) for n\in\mathbb{N}.
    (I guess the example with 3.14 might help you to see what to fill in.)
    intervals \langle n,n+1) ,  n \in \mathbb{N} ?
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  14. #14
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    Quote Originally Posted by jvignacio View Post
    intervals \langle n,n+1) ,  n \in \mathbb{N} ?
    Exactly.
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  15. #15
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    Quote Originally Posted by kompik View Post
    Exactly.
    mate thanks alot for the help. Really appreciate your time.
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