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Thread: getting started on a relation question

  1. #1
    Junior Member scottie.mcdonald's Avatar
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    Williamstown, Ontario, Canada
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    getting started on a relation question

    I just need help getting started on this question. the first line (or two) is eluding me:

    Let R be a relation on A and suppose that R is symmetric and transitive. Prove thefollowing:

    If, for all xE A, there is yE A such that xRy, then R is an equivalence relation.

    where xE A and yE A mean x and y are elements of A.

    I know i have to prove reflexitivity, and I only have this written down...my mind has stopped working:

    Let xE A, then R is an equivalence relation
    => (nothing written here...i'm stumped)

    any help would be appreciated. Thank you
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  2. #2
    Senior Member
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    Hi
    What you have to do is assume that $\displaystyle \forall x\in A,\ \exists y\in A\ \text{s.t.}\ x\mathcal{R}y$ and, with that hypothesis and the fact that $\displaystyle \mathcal{R}$ is symmetric and transitive, prove that $\displaystyle \mathcal{R}$ is an equivalence relation.

    As you said, you just have to prove that $\displaystyle \mathcal{R}$ is reflexive.

    So let $\displaystyle x$ be an element of $\displaystyle A,$ the hypothesis says that there is a $\displaystyle y\in A$ such that $\displaystyle x\mathcal{R}y.$
    But, since $\displaystyle \mathcal{R}$ is symmetric, what relation can we find using only that $\displaystyle x\mathcal{R}y.$ Finally, use the transitivity to prove that $\displaystyle x\mathcal{R}x.$
    Last edited by clic-clac; Feb 10th 2009 at 01:55 AM. Reason: b
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