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Homework Help: Reflexive, Symmetric, Transitive

  1. Mar 24, 2012 #1
    Indicate if the following relation on the given set is reflexive, symmetric, transitive on a given set.

    R where (x,y)R(z,w) iff x+z≤y+w on the set ℝxℝ.

    It is reflexive because any real number can make x+z=y+w.
    It is not symmetric because if x+z≤y+w it's not possible for x+z≥y+w.
    It is transitive

    Am I thinking about this correctly?
    Thank you
  2. jcsd
  3. Mar 24, 2012 #2
    To show something is reflexive, you need to show that a R a for all a in the set.

    So, does (x,y) R (x,y) for all real x,y?

    When showing symmetry, do not reverse the sign. For symmetry, we ask: Given (x,y)R(z,w), is (z,w)R(x,y)? Well, (x,y)R(z,w) implies that x+z<=y+w. And (z,w)R(x,y) implies that z+x<=w+y. So knowing that x+z<=y+w, is z+x<=w+y?
  4. Mar 24, 2012 #3
    (x,y)R(x,y) is true for all real x,y so the relation is reflexive.
    z+x≤w+y so (z,w)R(x,y) and the relation is symmetric.

    How would I show that the relation isn't transitive?
  5. Mar 24, 2012 #4


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    All you have done here is to write down the statement that you're supposed to prove or disprove. You also need to do the actual proof. Use the definition of R to find out if that statement is true or not.

    Why is z+x≤w+y? What are w,x,y,z anyway? If you're going to make a statement that involves a variable x, you need to do one of the following:

    1. Assign a value to x before you make the statement.
    2. Make it very clear that you're making a "for all x..." statement. (It's sufficient to say something like "let x be an arbitrary real number").
    3. Make it very clear that you're making a "there exists an x..." statement.

    Make a mental note of this. You need to apply this principle every time you try to prove something.

    Find a counterexample.
    Last edited: Mar 24, 2012
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