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Thread: reflexive, symetric,transitive - examples?

  1. #1
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    reflexive, symetric,transitive - examples?

    Hello,
    I was wondering if someone could help me with this problem:

    A relation on set can be reflexive (R), symmetric (S) or transitive(T) (..of course ) Depending on which of these ''basic characteristics'' it has, there are 8 options:
    1. R, S, T
    2.only R
    3.only S
    4.only T
    5. R,S
    6.R,T
    7.S,T
    8. neither R nor S nor T
    I have to find an example for each option. I guess it's very easy, but somehow I have difficulties finding them, especially the ones that have only one characteristic. The first one I have of course : ) Any help is appreciated!
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Nice question
    I know some

    1.a~b as "$\displaystyle a=b$" (any equivalence relation works)
    2.
    3a~b as "a.b = a+b"
    4 a~b as "$\displaystyle a< b$"
    5
    6a~b as " $\displaystyle a\subset b$ " or "a|b" or "$\displaystyle \leq $"
    7a~b as "$\displaystyle \max\left\{a,b\right\}< a+b$" (~ on Z for example)
    8 a~b as $\displaystyle a+1 = b$

    Ill try to think of more
    Last edited by Dinkydoe; Dec 30th 2009 at 06:11 AM.
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  3. #3
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    Quote Originally Posted by Dinkydoe View Post
    Ill try to think of more
    Thanks for your help!
    Hmm do you think that an example for 2. could be:
    (x,x); x∈X (as the diagonal, the smallest reflexive relation on X )
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  4. #4
    Senior Member Dinkydoe's Avatar
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    I'm not so sure. Then the relation is certainly reflexive, but probably symmetric and transitive too. "if" a~b then b~a since a=b in this case.

    I hope you've had some topology:

    Taking the standard topology on R: Let A~B <-->A\B is open
    Reflexivity: A\A = $\displaystyle \emptyset $ is open.

    Not symmetric: Let A = (0,1), B={2} then A\B = A is open. B\A= B is closed.
    Not transitive: Let A = (0,1) and B=(2,3), C= (0,1/2) Then A\B=(0,1) is open, B\C=(2,3) is open. But A\C = [1/2,1) is not open.

    There must be very simple examples, but I can't find one.
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  5. #5
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    Hello, Dandelion!

    A relation on set can be reflexive (R), symmetric (S) or transitive(T).

    There are 8 options:

    . . $\displaystyle \begin{array}{c| ccc}1&R&S&T \\2&R&-&- \\ 3.&-&S&- \\ 4.&-&-&T\\ 5.&R&S&- \\ 6.&R&-&T \\ 7.&-& S&T \\ 8.&-&-&-
    \end{array}$


    I have to find an example for each option.
    I guess it's very easy. . .
    no, it isn't!
    This really stretches one's imagination.



    $\displaystyle 1.\;R,S,T$
    Any relation involving "equality".
    Examples: "is equal to", "is as old as", "is as tall as", etc.



    $\displaystyle 3.\:S\text{ only}$
    $\displaystyle "X\text{ is married to }Y."$
    Assume this involves a legal ceremony.

    Not reflexive: .$\displaystyle a$ is not married to him/herself.

    Symmetric: .$\displaystyle (a\circ b) \:\to\: (b\circ a)$

    Not transtive: .$\displaystyle \bigg[(a\circ b) \wedge (b \circ c)\bigg] \;\rlap{\;\;/}\to\;(a\circ c)$
    . .
    unless, of course, bigamy is legal on your planet.



    $\displaystyle 4.\;T\text{ only}$
    Any relation involving "inequality".
    Examples: "is older than", "is shorter than", etc.



    $\displaystyle 5.\;R\text{ and }S\text{ only}$
    "$\displaystyle X$ lives within one mile of $\displaystyle Y.$"

    $\displaystyle a\circ a:\;\;a$ lives within one mile of him/herself (reflexive).

    $\displaystyle \text{If } a\circ b\text{, then }b\circ a:\;\;a\text{ and }b$ live within one mile of each other (symmetric).

    Not transitive: .$\displaystyle a\circ b \text{ and }b\circ c $ does not imply $\displaystyle a \circ c$
    . . . $\displaystyle a$ and $\displaystyle b$ can be a mile apart.
    . . . $\displaystyle b$ and $\displaystyle c$ can be a mile apart.
    But $\displaystyle a$ and $\displaystyle c$ can be two miles apart.



    $\displaystyle 6.\;R\text{ and }T\text{ only}$
    $\displaystyle "X$ is a brother of $\displaystyle Y."$
    Assume that this means: $\displaystyle X$ is male and has the same parents as $\displaystyle Y.$

    $\displaystyle a\circ a:\;\; a$ is a brother of himself (reflexive).

    $\displaystyle \text{If }a\circ b\text{ and }b\circ c\text{, then }a\circ c\!:$ (transitive)

    Not symmetric: .$\displaystyle a\circ b$ does not imply $\displaystyle b \circ a.$
    . . $\displaystyle a$ can be a brother of $\displaystyle b$, but $\displaystyle b$ can be a sister of $\displaystyle a.$



    $\displaystyle 8.\;\sim\!R,\:\sim\!S,\:\sim\!T $
    $\displaystyle "X\text{ is the father of }Y."$

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  6. #6
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    Quote Originally Posted by Dandelion View Post
    A relation on set can be reflexive (R), symmetric (S) or transitive(T) (..of course ) Depending on which of these ''basic characteristics'' it has, there are 8 options:
    1. R, S, T
    2.only R
    3.only S
    4.only T
    5. R,S
    6.R,T
    7.S,T
    8. neither R nor S nor T
    While the replies are correct, there is an easier way.
    Let $\displaystyle A=\{p,q,r\}$ then $\displaystyle A \times A$ answers #1.

    The diagonal $\displaystyle \Delta_A=\{(p,p),(q,q),(r,r)\}$ is reflexive but it is also transitive and symmetric. So it also satisfies #1.

    The relation $\displaystyle \Delta_A\cup\{(p,q),(q,r)\}$ is reflexive but is neither symmetric nor transitive.

    Using this simple set we can continue to find examples for all eight.
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  7. #7
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    Thank you all for your help and kindness!
    I think I can definitely write the answer now. I might try to combine those different methods, although, I must admit, I didn't know we could use human relations as a mathematical example =)

    Anyway, thanks once again and Happy New Year!!
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