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The question asks if there can be a relation on a set that is neither reflexive or irreflexive. The example the book give makes perfect sense: "Yes, for instance{(1,1)} on{1,2}."

I was wondering, if I had the relation on that same set {(1, 2)}, would that be irreflexive?

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Yes. In fact, the relation $\{\langle 1,2\rangle\}$ is irreflexive on every set that contains both $1$ and $2$: it contains no ordered pairs of the form $\langle x,x\rangle$. –  Brian M. Scott Nov 3 '12 at 17:14
Thank you very much. That was an insightful response. –  Mack Nov 3 '12 at 17:22
You’re very welcome. –  Brian M. Scott Nov 3 '12 at 17:23

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