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Thread: Relations

  1. #1
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    Relations

    Prove:

    $\displaystyle R\circ (S\cap T)\subseteq R \circ S \cap R \circ T $

    and an example where inclusion is strict

    $\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$

    and an example where inclusion is strict

    I'm not understanding the composition of the intersection
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  2. #2
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    Quote Originally Posted by gutnedawg View Post
    Prove:
    $\displaystyle R\circ (S\cap T)\subseteq R \circ S \cap R \circ T $

    and an example where inclusion is strict
    $\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$
    and an example where inclusion is strict
    $\displaystyle R\circ (S\cap T)\subseteq (R \circ S) \cap (R \circ T) $
    Written that way the statement is true.
    I have no idea what $\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$ could mean.
    Have you miss-written that?
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  3. #3
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    I know it is true I need to prove it and no I did not miss write the last part
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  4. #4
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    Please answer both of my questions.
    The second one you wrote is meaningless.
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  5. #5
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    I have prove one of the modular laws for realtions R, S, T

    $\displaystyle R \circ S \cap T \subseteq R \circ (S\cap R^-^1 \circ T)$

    or

    $\displaystyle R \cap S \circ T \subseteq (R\circ T^-^1 \cap S) \circ T$
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  6. #6
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    Quote Originally Posted by gutnedawg View Post
    I have prove one of the modular laws for realtions R, S, T
    $\displaystyle R \circ S \cap T \subseteq R \circ (S\cap R^-^1 \circ T)$
    or $\displaystyle R \cap S \circ T \subseteq (R\circ T^-^1 \cap S) \circ T$
    The way both of those are written makes totally meaningless.
    Look at the way I used parentheses in post #2.
    Without parentheses there is no way to know what goes with what.

    In the second one it makes no sense to have $\displaystyle R^{-1}$ in it.
    If you cannot provide a readable question, the we cannot help.


    Now here is the proof of a standard question.
    If $\displaystyle (a,b) \in R \circ \left( {S \cap T} \right)$ then $\displaystyle \left( {\exists c} \right)\left[ {(a,c) \in \left( {S \cap T} \right)\;\& \;(c,b) \in R} \right]$.

    That also means $\displaystyle \left[ {\left( {(a,c) \in S\;\& \;(c,b) \in R} \right)\;\& \;\left( {(a,c) \in T\;\& \;(c,b) \in R} \right)} \right]$.

    Or $\displaystyle (a,b) \in \left( {R \circ S} \right) \cap \left( {R \circ T} \right)$.
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  7. #7
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    $\displaystyle R^-^1 $

    is the inverse relation
    $\displaystyle \{(b,a) : (a,b) \epsilon R\}$
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