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I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood.

Anselm
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6 Answers6

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For any equivalence relation $\sim$, $x\nsim y$.

This captures the example of "equality" that people came up with earlier, and grabs other similar things like "is isomorphic to", etc.

Strictly speaking, you are not using transitivity at all, so any reflexive symmetric relation would do.

There are natural examples of symmetric, reflexive, nontransitive relations. One is using a distance relation for points in the plane: $x\sim y$ iff $d(x,y)<1$. So, $x\nsim y$ for this relation is an example different from $\neq$ :)

Finally, you can always just engineer one by hand. Let $X=\{a,b\}$. Then the relation $\{(x,y),(y,x)\}$ is symmetric but not reflexive. (I consider empty relations to be cheating :) )

rschwieb
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    +1 I recall the math.se post with examples of symmetric, reflexive, and non-transitive relations quite well ;-) – amWhy May 08 '13 at 20:49
  • @amWhy I wasn't even aware of it. Here it is for the curious: http://math.stackexchange.com/questions/268726/are-there-real-life-relations-which-are-symmetric-and-reflexive-but-not-transiti . It's no wonder you haven't forgotten it: I would not forget an answer of mine with $12^2$ (and counting) upvotes either :) – rschwieb May 08 '13 at 20:59
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Over say the integers: $x$ is not equal to $y$.

You can play this game with most mathematical notions: not congruent, if you like geometry, not of the same cardinality, if you want set theory, and so on.

André Nicolas
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Take $\{(x,y)\in \mathbb Z^2: x = -y\}$

John Douma
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Take the set $A:=\{0,1\} $ (someone might just call it 2) and consider the relation on $A$ given by $R:=\{(0,1),\ (1,0)\}$: it is trivially symmetric but not reflexive.

Marco Vergura
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Here is an example that isn't really in the spirit of the question, but I'll provide it anyway.

Let $R$ be any symmetric relation (possibly even reflexive) on a set $X$. Define a new relation $R^\prime$ by $$ R^\prime = R \setminus \{(x,x) \mid x \in X\}. $$

What we've done is take a symmetric relation and simply remove all the reflexive bits. (In fact, we could have just removed a single pair $(x,x)$, since a single counterexample is enough to destroy reflexivity.)

The purpose of my example is to emphasize that, when constructing mathematical objects, you shouldn't feel restricted to the usual suspects. Feel free to really force the desired behavior explicitly.

Austin Mohr
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Let $X=\{a,b\}$ be a set, ($a$ and $b$ distinct). Define the relation $R$ on X by $R=\{(a,a)\}$. Then $R$ is symmetric (and transitive), but not reflexive on $X$ since $(b,b)$ is not in $R$.

Federica Maggioni
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