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Transitive Set Question

  1. Apr 28, 2014 #1
    1. The problem statement, all variables and given/known data

    a) Consider the relation S de fiend on the set {t : t is a person} such that xSy holds exactly if
    person x is taller than y. Determine if the relation S is reflexive, symmetric and transitive.
    Is the relation S an equivalence relation?

    2. Relevant equations
    Recall that a relation R de ned on a set A is reflexive if for all x 2 A xRx.
    Recall that a relation R de ned on a set A is symmetric if for all x 2 A and y 2 A the xRy implies
    yRx.
    Recall that a relation R e ned on a set A is transitive if for all x; y; z in A, both xRy and yRz
    holds, then xRz holds as well.
    Finally recall that a relation R is an equivalence relation if its reflexive, symmetric and transitive.


    3. The attempt at a solution

    As far as i can see the set is not symmetric or reflexive but i'm not 100% on transitive...
    It would be transitive if x > y and y > z then x > z holds...but we aren't given any information on y > z?

    So would this set be transitive?
     
  2. jcsd
  3. Apr 28, 2014 #2

    Zondrina

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    Homework Helper

    Yes the relationship is not symmetric or reflexive, but it IS transitive.

    If ##xRy## holds, then ##x > y##, if ##yRz## holds, then ##y > z##. So person ##x## is taller than person ##y## and person ##y## is taller than person ##z##.

    It must be the case that person ##x## must be taller than person ##z##.
     
  4. Apr 28, 2014 #3
    Yes, i agree with what you said...but the question does not mention that y > z
     
  5. Apr 28, 2014 #4

    Zondrina

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    Homework Helper

    According to your definition (with some slight corrections):

    Notice I added the word 'if'. If you assume they hold, does the conclusion still hold?
     
  6. Apr 28, 2014 #5
    Yeah then it holds. The definition is provided as part of the information the exam question but i'll just assume it in the exam as well. Thanks
     
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