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Math Help - equivalence relation

  1. #1
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    equivalence relation

    The relation R is defined by aRb if ab=ba for all a,b in Z(set of integers).

    How to show that R is not an equivalence relation?
    -------------
    R is definitely reflexive and symmetric.
    How to prove that it is not transitive
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  2. #2
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    Re: equivalence relation

    Quote Originally Posted by Suvadip View Post
    The relation R is defined by aRb if ab=ba for all a,b in Z(set of integers).
    How to show that R is not an equivalence relation?
    -------------
    R is definitely reflexive and symmetric.
    How to prove that it is not transitive
    OH? How do you define 0^0~?
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  3. #3
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    Re: equivalence relation

    Hello, Suvadip!

    This is a strange problem.
    I would like to see the textbook solution.


    The relation R is defined by: . aRb\text{ if }a^b =b^a\text{ for all }a,b \in Z.

    How to show that R is not an equivalence relation?

    - - - - - - - - - - - - - - - - - - - - - - - - -

    R is definitely reflexive and symmetric.

    How to prove that it is not transitive? .I don't know . . .

    Note that: a,b \ne0


    Does aRb and bRc imply aRc\,? .Yes!

    \begin{array}{ccccccc}aRb & \Rightarrow & a^b \:=\:b^a & [1] \\ bRc & \Rightarrow & b^c \:=\:c^b & [2] \end{array}

    Raise [1] to the power \tfrac{c}{b}\!:\;\left(a^b\right)^{\frac{c}{b}} \:=\:\left(b^a\right)^{\frac{c}{b}} \quad\Rightarrow\quad a^c \:=\:b^{\frac{ac}{b}}\;\;[3]

    Raise [2] to the power \tfrac{a}{b}\!:\;\left(b^c\right)^{\frac{a}{b}} \:=\:\left(c^b\right)^{\frac{a}{b}} \quad\Rightarrow\quad b^{\frac{ac}{b}} \:=\:c^a\;\;[4]

    Equate [3] and [4]: . a^c \:=\:c^a

    Therefore, R is transitive. . R is an equivalence relation.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    One explanation . . .

    We can find one such relation: . 2R4 \quad\Rightarrow\quad 2^4 \:=\:4^2

    The difficulty is finding the second relation: . 4Rc \quad\Rightarrow\quad 4^c \:=\:c^4
    . . where c \ne 2.
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  4. #4
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    Re: equivalence relation

    Quote Originally Posted by Plato View Post
    OH? How do you define 0^0~?
    R cannot be reflexive as aRa does not hold for a=0 ( since 0^0 is not defined ). Am I right?
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  5. #5
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    Re: equivalence relation

    Quote Originally Posted by Suvadip View Post
    R cannot be reflexive as aRa does not hold for a=0 ( since 0^0 is not defined ). Am I right?
    That strictly depends upon who you think is correct. Study this page.

    There is no general agreement on that point. So you must go by whatever your textbook/lecturer says.

    For me 0^0 is not defined.
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