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Proving irreflexive and symmetric relation

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data
    I am on the final part of a question and I have to prove that the following is a irreflexive symmetric relation over A or if it is not then give a counter example.

    R is given as an irreflexive symmetric relation over A.

    Relation: {(X, Y) | X ⊆ A ∧ Y ⊆ A ∧ ∀x ∈ X.∀y ∈ Y.(x, y) ∈ R}

    2. Relevant equations
    See above.


    3. The attempt at a solution
    I have worked out the if X x Y ⊆ R then (X,Y) is put into the relation.
    I worked out a simple example to see if it was worth trying to prove and it seems to be correct.
    I am having trouble getting my head around trying to make a start to prove this.

    Many thanks.
     
  2. jcsd
  3. Apr 9, 2012 #2
    Irreflexive means not reflexive, yes? Then a possible counter-example might exist when X and Y are not disjoint. Wouldn't it?
     
  4. Apr 10, 2012 #3
    I think your counterexample is incorrect.
    R needs to be irreflexive, so if two sets have an element in common, R is not irreflexive and so the sets don't belong to the relation.
     
  5. Apr 10, 2012 #4
    Let's call your relation J.

    To prove the symmetric part. Assume X J Y, this means
    X ⊆ A ∧ Y ⊆ A ∧ ∀x ∈ X.∀y ∈ Y.(x, y) ∈ R and (X,Y) belongs to J
    use the fact that R is symmetric to arrive at
    Y ⊆ A ∧ X ⊆ A ∧ ∀y ∈ Y.∀x ∈ X.(y, x) ∈ R and (Y,X) belongs to J
    which means Y J X

    To prove the irreflexive part i would go for a contradiction. Since the statement you have to prove is a negative one. Which is: "for any set X, (X,X) does not belong to J". So assume the contrary and reach a contradiction.
     
  6. Apr 10, 2012 #5
    Thank you for the help. I think I was too worried about the complexity of the set rather than using the definition to simply prove it. I should be able to move on to some more complex ones now.

    Thanks again.
     
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