Suppose you take the relation R containing just the ordered pair <1,0>. Can this be a relation on the empty set?

One line of reasoning might be that R is an equivalence relation on the empty set, because there is nothing in the empty set that could make R irreflexive, etc.

But on the other hand, how can R be a subset of the Cartesian product of the empty set, given that R itself is not empty but contains the ordered pair <1,0>?

I feel I'm confused somewhere and would appreciate your help!

Thanks :@)

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    $\begingroup$ A relation on a set $S$ is a set of ordered pairs $\langle x,y\rangle$ with $x,y\in S$. No ordered pair can have its elements in the empty set. In other words, the only relation on the empty set is the empty relation. $\endgroup$ – egreg Oct 6 '14 at 22:22
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    $\begingroup$ A relation on a set $X$ is a subset of $X\times X$. As $\{\langle 1,0 \rangle\}$ is not a subset of $\varnothing\times \varnothing$ then it is not a relation on $\varnothing$. $\endgroup$ – James Oct 6 '14 at 22:22

A relation on $S$ is a set of ordered pairs $\langle x,y\rangle$ with $x,y\in S$ or, said in a more precise way, is a subset of $S\times S$.

Since $\emptyset\times\emptyset=\emptyset$, the only relation on $\emptyset$ is the empty relation.

This empty relation on $\emptyset$ is reflexive, antireflexive, symmetric, antisymmetric and also transitive.

So, no, the relation $\{\langle 1,0\rangle\}$ is not a relation on $\emptyset$, but it is a relation on any set containing $\{0,1\}$.

  • $\begingroup$ Thank you for your answer, that's a clear way of setting it out. So would it be simply wrong to say, e.g., that any relation on the empty set is reflexive? I've heard this claim before, but it seems incompatible with saying that the null relation is the only relation on the empty set. The argument that is sometimes presented is that 'R is reflexive on S' should be taken to mean 'there is no element d in S for which <d,d> is not in relation R', which would be trivially true for the empty set. $\endgroup$ – Tweedle Dum Oct 6 '14 at 23:40
  • $\begingroup$ @TweedleDum Yes, any relation (but there's only one) on the empty set is reflexive. The statement “If $(x,y)\in\emptyset$ then $(y,x)\in\emptyset$” is certainly true, because “$(x,y)\in\emptyset$” is false. $\endgroup$ – egreg Oct 7 '14 at 6:21

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