Properties of the relation {<1,0>} on the empty set??

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 :@)

• 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. – egreg Oct 6 '14 at 22:22
• 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 $\varnothing$. – 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 $\{0,1\}$.
• @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. – egreg Oct 7 '14 at 6:21