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Math Help - Equivalence Relation and Classes

  1. #1
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    Equivalence Relation and Classes

    Hello, I have the following problem, I think I've proved the equivalence relaltion correctly, but cannot figure out how to work the equivalence classes. Hopefully some one will know!

    The relation T is defined on the set of integers by:

    aTb if and only if ab>0 or a=b=0

    For the equivalence realtion I have come up with the following

    1. Reflexive: aTa as a=a=0 or we could have stated that aa>0

    2. Symmetric: Take integers a,b. aTb means ab>0 or a=b=0 => b=a=0 or ba>0. Hence we also have bTa

    3. Transitive: Take integers a,b,c. If we have aTb and bTc, ab>0 or a=b=0 and bc>0 or b=c=0. So we have a=b=c=0, hence, bTc

    Then, the next question I was asked was what are the equivalence relations?
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  2. #2
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    Quote Originally Posted by iwish123 View Post
    The relation T is defined on the set of integers by:
    aTb if and only if ab>0 or a=b=0

    For the equivalence realtion I have come up with the following
    1. Reflexive: aTa as a=a=0 or we could have stated that aa>0
    2. Symmetric: Take integers a,b. aTb means ab>0 or a=b=0 => b=a=0 or ba>0. Hence we also have bTa
    3. Transitive: Take integers a,b,c. If we have aTb and bTc, ab>0 or a=b=0 and bc>0 or b=c=0. So we have a=b=c=0, hence, bTc
    Then, the next question I was asked was what are the equivalence relations?
    Note that n^2\ge 0 that is proves the relation is reflexive.

    The suppose that aTb~\&~bTc you want to show aTc.
    Start with if b=0 then at once that means that a=0=c done.
    On the other hand, if b\not= 0 then a,~b,~\&~c all have the same sign. So ac>0.

    There are only three equivalence classes. One is the set \{0\}.
    Now what are the other two?
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  3. #3
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    Thank you for the help, would the other 2 equivalence classes be numbers greater than 0 and numbers less than 0? In what format would I write that in exam if im correct?
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  4. #4
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    Quote Originally Posted by iwish123 View Post
    the other 2 equivalence classes be numbers greater than 0 and numbers less than 0?
    Correct!

    Quote Originally Posted by iwish123 View Post
    In what format would I write that in exam if im correct?
    I have no idea. That strictly depends on your textbook/notes/instructor.
    There are many notations for equivalence classes in use.
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  5. #5
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    Thanks again.
    I've practiced another equivalence relation question involving complex numbers.

    The relation R is defined for complex numbers z=x+yi and w=a+bi. zRw if and only if x+b=y+a

    I am asked to give the elements of the equivalence class containg [i]

    I have wrote, [i]={xeC:xRi}={xeC:x-i}, so would the elements of the equivalence class [i] be those numbers congruent to i or -i, so the elements would be 1 and -1?
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