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In the case below, a relation on the set $\{1, 2, 3\}$ is given. Of the three properties, reflexivity, symmetry, and transitivity, determine which ones the relation has. Give reasons.

$R = \emptyset$

I guessed that it was reflexive, symmetric, and transitive. My teacher said that it was not reflexive, but I didn't understand why. My reasons for it being reflexive, symmetric, and transitive was because they didn't have an if part that could make it false. Is this the same as saying that it is vacuously true? If not, what would be the reasons for whether or not it is reflexive, symmetric, and transitive?

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To be reflexive on $S$, you would need to have $(i,i) \in R$, for $i \in \{1,2,3\}$. – copper.hat Mar 3 '13 at 8:23
up vote 5 down vote accepted

Being a reflexive relation is an extrinsic property. It connects the relation with an external set, while being symmetric and transitive are intrinsic properties that depend only on the ordered pairs in the relation.

$R$ is reflexive on $A$ if for every $x\in A$ the pair $\langle x,x\rangle$ is in $R$. So if $R=\varnothing$ and $A\neq\varnothing$ it is not reflexive on $A$.

The same would hold whether $\{\langle 1,1\rangle\}$ is reflexive on $\{1\}$? Yes. It is, but is it reflexive on $\Bbb N$? No. The pair $\langle 2,2\rangle$ is not there, so not every $n\in\Bbb N$ satisfies the property required.

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