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Construct a smallest possible relation on the set {a, b, c, d} that is irreflexive, symmetric, and transitive. I thought that the relation would be an empty set, R is symmetric because (a,b) can never belong to it and so the conditional statement is always true. R is transitive because (a,b) and (b,c) can never belong to it and so the conditional statement is always true. do you think that there is anything wrong with my assumption?

RobPratt
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louisd
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  • https://math.stackexchange.com/questions/440/why-isnt-reflexivity-redundant-in-the-definition-of-equivalence-relation – Xander Henderson Apr 07 '20 at 21:45
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    See also [1], [2], and [3]. There are other examples which can be found by searching this site for the terms "symmetric transitive irreflexive" and/or variations on that theme. – Xander Henderson Apr 07 '20 at 21:48
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    @XanderHenderson: your link [2] is the one that actually provides an answer to this question. I don't think your other links are pertinent. – Rob Arthan Apr 07 '20 at 22:17
  • The duplicate addresses relations that are not reflexive (meaning there is at least one element $a$ in the set such that $(a, a) \notin R$. However, if by irreflexive you mean that there is *no* $x \in {a, b, c, d}$ such that $(x, x) \in R$, then you are correct that such a relation can only be the empty relation $R = { } = \varnothing.$ – amWhy Apr 07 '20 at 22:36

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Note, that there are in fact, non-empty relations that are not reflexive, but are symmetric and transitive..

A relation $R$ that is not reflexive on a set $\{a, b, c\}$ means simply that one or more of $(a, a), (b, b), (c, c) \notin R$.

So, for example, $R_1 = \{(a, a), (a, b), (b, a), (b, b)\}$ is not reflexive (because $(c, c) \notin R_1$), but it is symmetric and transitive.

However, you stipulated in your question that you were wondering what a relation on a set could be that is irreflexive, symmetric, and transitive would be.

A relation $R_2$ that is irreflexive on a set $A$ is a relation for which there is no element $x\in A$, for which $(x, x) \in R$. So, for example, in our previous example of a non-reflexive relation, we'd have to remove $(a, a), (b, b) \in R_2$. If we remove $(a, a)$ and $(b, b)$ from $R_2$, then it ceases to be transitive, because we'd have $(a, b) \in R_2$, and $(b, a) \in R_2$ but not $(a, a), (b, b)\in R_2$. So we'd need to remove $(a, b), (b, a)$ from $R_2$. Hence we are left with an empty set, $R_2 = \varnothing.$

So the only relation that is irreflexive, symmetric, and transitive, on any set, is in fact, the empty set.

amWhy
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