Matrix relation of sets. symmetric, antisymmetric,reflexive,transitive

  1. 1. The problem statement, all variables and given/known data

    relation A = {a,b,c} for the following matrix [1,0,0;1,1,0;0,1,1]

    is it reflexive, transitive, symmetric, antisymmetric


    2. Relevant equations

    ordered pairs.

    3. The attempt at a solution

    i wrote the ordered pairs as (a,a),(b,a),(b,b),(c,b),(c,c)

    I only that it is reflexive for a,a b,b and c,c
    also it is antisymmetric because there are no edges in opposite directions between distinct verticies.

    am I missing anything. thanks!
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,391
    Staff Emeritus
    Science Advisor

    I don't know what you mean by "reflexive for a,a b,b and c,c. A relation is reflexive if and only if it contains (x,x) for all x in the base set. Since only a, b, and c are in the base set, and the relation contains (a,a), (b,b), and (c,c), yes, it is reflexive.

    To be symmetric, since it contains (b,a) it would have to contain (a,b) and it doesn't: not symmetric. Since it does NOT contain (a,b) or (b,c), yes, it is anti-symmetric.

    What about transitive? A relation is transitive if and only if, whenever (x,y) and (y,z) are in the relation, so is (x,z). Can you find pairs so that is NOT true?
     
  4. Hey, thanks for the reply!

    I didn't put parenthesis around the ordered pairs (a,a),(b,b),(c,c) for the first problem, sorry.

    I don't think it's transitive since we have (c,b) and (b,a), and it doesn't contain (c,a). How does that sound? Thanks
     
  5. HallsofIvy

    HallsofIvy 40,391
    Staff Emeritus
    Science Advisor

    Yes, that completes it.
     
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