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Math Help - Prove or disprove that R^4 is a transitive relation

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    Prove or disprove that R^4 is a transitive relation

    Let R be a transitive relation defined on a set B. Prove or disprove that R^4 is a transitive relation.

    I'm inclined to believe that it's true, but I can't be sure. I at least know this: A relation R is transitive on a set A if whenever (a,b)\in R and (b,c)\in R, then (a,c)\in R, \forall a,b,c\in R
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  2. #2
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    Suppose that R is transitive on A.
    Suppose that  \left[ {\left( {a,b} \right) \in R \circ R} \right] \wedge \left[ {\left( {b,c} \right) \in R \circ R} \right].
    By definition \left( {\exists x \in A} \right)\left[ {(a,x) \in R \wedge (x,b) \in R} \right] \wedge \left( {\exists y \in A} \right)\left[ {(b,y) \in R \wedge (y,c) \in R} \right].
    But that means that \left[ {(a,b) \in R \wedge (b,c) \in R} \right] because Ris transitive.
    So by way of b we have {\left(a,c \right) \in R \circ R}.
     R \circ R is transitive.
    Can you extend that to R^4?
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