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Math Help - Properties of Relations

  1. #1
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    Exclamation Properties of Relations

    I'm having real trouble being able to identify when a relation is reflexive, irreflexive, symmetric, antisymmetric, and transitive.

    For instance, here is a relation on the set {1, 2, 3, 4, 5}

    R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

    What exactly do you look for when determining if the set has each quality mentioned above? I know the basics (i.e. reflexive is for all a, aRa) but I seem to be getting these wrong each time. If someone could map out how they approach this problem in their mind I would really appreciate it.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by brand_182 View Post
    I'm having real trouble being able to identify when a relation is reflexive, irreflexive, symmetric, antisymmetric, and transitive.

    For instance, here is a relation on the set {1, 2, 3, 4, 5}

    R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

    What exactly do you look for when determining if the set has each quality mentioned above? I know the basics (i.e. reflexive is for all a, aRa) but I seem to be getting these wrong each time. If someone could map out how they approach this problem in their mind I would really appreciate it.
    Here, use _aR_b to mean (a,b) \in R.

    The relation R is reflexive if (a,a) \in R for all a \in \{ 1,2,3,4,5 \}

    The relation is irreflexive if (a,a) \not \in R for all a \in \{ 1,2,3,4,5 \}

    The relation is symmetric if (a,b) \in R implies (b,a) \in R, for all a,b \in \{ 1,2,3,4,5 \}

    The relation is antisymmetric if (a,b) \in R and (b,a) \in R implies a = b, for all a,b \in \{ 1,2,3,4,5 \}

    The relation is transitive if (a,b) \in R and (b,c) \in R implies (a,c) \in R, for all a,b,c \in \{ 1,2,3,4,5 \}


    The important word is "all," you must check that these definitions hold for all elements in R
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    thank you very much!
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