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Math Help - relations transitive

  1. #1
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    relations transitive

    How come 2) is transitive and not 1)

    1) R_1 = (1,1) (1,2) (2,1)
    2) R_2 = (2,1) (3,1) (3,2) (4,1) (4,2) (4,3)

    Definition of transitive relation states that for every(a,b,c) belongs to R
    (a,b)belongs to R AND (b,a)belongs to R then (a,c) belongs to R.

    So how come 1) is NOT transitive and 2) IS transitive?

    Can anyone clarify my doubt!?

    Thank you.
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  2. #2
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    Hello, robocop_911!

    1)\;\;R_1 \:= \:\{(1,1),\:(1,2),\:(2,1)\}
    2)\;\; R_2 \:=\:\{ (2,1),\: (3,1),\: (3,2),\: (4,1),\: (4,2),\: (4,3)\}

    Definition of transitive relation states that:
    for every a,b,c \in R,\;(a,b) \in R\,\text{ AND }\,(b,c) \in R, then (a,c) \in R.

    So how come 1) is NOT transitive and 2) IS transitive?

    In R_1, we have:

    (1,1) \in R_1 \:\wedge \:(1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1 . . . True

    (1,2) \in R_1 \:\wedge \:(2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1 . . . True

    (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1 . . . False!

    . . R_1 is not transitive.



    In R_2, we have:

    (4,2) \in R_2 \:\wedge \:(2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2 . . . True

    (4,3) \in R_2 \:\wedge \:(3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2 . . . True

    . . R_2 is transitive.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, robocop_911!


    In R_1, we have:

    1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1" alt="(1,1) \in R_1 \:\wedge \1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1" /> . . . True

    2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1" alt="(1,2) \in R_1 \:\wedge \2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1" /> . . . True

    (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1 . . . False!

    . . R_1 is not transitive.



    In R_2, we have:

    2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" alt="(4,2) \in R_2 \:\wedge \2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" /> . . . True

    3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" alt="(4,3) \in R_2 \:\wedge \3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" /> . . . True

    . . R_2 is transitive.

    I disagree...
    (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1 . . . False!

    . . R_1 is not transitive.

    it could bye (2,1) ^ (1,1) => (2,1) and (2,1) is in the set!
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  4. #4
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    Recall that transitive property says for all elements a,b,c a~b, b~c => a~c
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  5. #5
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    Quote Originally Posted by Isomorphism View Post
    Recall that transitive property says for all elements a,b,c a~b, b~c => a~c
    I am considering All the elements! Still, I am not getting the correct answer.
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  6. #6
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    How come only...

    R_6 = (3,4) is Also transitive? This is really mindboggling!
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