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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
This question appears to be off-topic because it is essentially a yes-or-no question with no conceivable future value. – Lord_Farin Dec 25 '14 at 22:11
@k170, did you really consider that bit of MathJax was enough justification to bump this very narrowly scoped, two-year-old post I was trying to purge from the system? Please look at the last active date before editing. – Lord_Farin Dec 25 '14 at 22:42

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