1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook