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Math Help - Relations

  1. #1
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    Relations

    Prove:

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

    and an example where inclusion is strict

    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:
    R\circ (S\cap T)\subseteq R \circ S \cap R \circ T

    and an example where inclusion is strict
    R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)
    and an example where inclusion is strict
    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 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

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

    or

    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
    R \circ S \cap T \subseteq R \circ (S\cap R^-^1 \circ T)
    or 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 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 (a,b) \in R \circ \left( {S \cap T} \right) then  \left( {\exists c} \right)\left[ {(a,c) \in \left( {S \cap T} \right)\;\& \;(c,b) \in R} \right].

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

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

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