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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 $\langle x,x\rangle$.2012-11-03
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    Thank you very much. That was an insightful response.2012-11-03
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    You’re very welcome.2012-11-03
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    This question appears to be off-topic because it is essentially a yes-or-no question with no conceivable future value.2014-12-25
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    @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.2014-12-25
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    @Lord_Farin, forgive me my Lord. I saw it on the review queue and I added MathJax irrespective of the date. Feel free to vote to close.2014-12-25
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    @k170 Please read [this](http://meta.stackexchange.com/a/151443/222340); your actions have undesirable consequences. In the future, please only click edit if you think the edit will make the question worthy of staying open.2014-12-25
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    @Lord_Farin, thanks for the link. I'll be more careful next time.2014-12-25

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