Discrete Mathematics – Proving that the empty relation on a non-empty set is asymmetric

discrete mathematicselementary-set-theoryproof-verificationrelations

Prove that the empty relation, defined on a non-empty set $A$, is asymmetric.

My work:

We need to show that if $\mathcal{R}=\varnothing$ is a relation on $A\times A$, with $A\neq\varnothing$, then for all $x,y\in A$, if $(x,y)\in\mathcal{R}$ then $(y,x)\notin\mathcal{R}$.

Proof Since $A\neq\varnothing$ let $x,y\in A$. Since $\mathcal{R}\subseteq A\times A$ then $(x,y)\in R$. By hypothesis, $(x,y)\in\varnothing$, but this is not true, because the empty relation has no element, so we have $\mathrm{F}\to\text{?}$ thus $(y,x)\notin\mathcal{R}$ (false antecedent implies that the conditional is true), hence $\mathcal{R}$ is asymmetric.

Is it correctly justified and is my reasoning consistent?

Best Answer

Your proof has been pretty severely critiqued in the comments, so I'll compare it to an exemplar:

Since $A\neq\varnothing$ let $x,y\in A$. Since $\mathcal{R}\subseteq A\times A$ then $(x,y)\in R$. By hypothesis, $(x,y)\in\varnothing$, but this is not true, because the empty relation has no element, so we have $\mathrm{F}\to\text{?}$ thus $(y,x)\notin\mathcal{R}$ (false antecedent implies that the conditional is true), hence $\mathcal{R}$ is asymmetric.

Let $x,y\in A$ be given. Since $\mathcal R$ is empty, $(x,y)\in\mathcal R$ is false. Therefore, $(x,y)\in \mathcal R\to(y,x)\notin\mathcal R$ is vacuously true. Since $x$ and $y$ were arbitrary, $\mathcal R$ is asymmetric.

  • Since you are letting $x$ and $y$ be arbitrary members of $A$ instead of choosing them from $A$, you do not need to observe that $A$ is non-empty. (In fact, the empty relation over the empty set is also asymmetric.)

  • Your statement that $(x,y)\in\mathcal R$ is poorly reasoned, and vacuously false. If you meant to say that that statement is well-formed and therefore a statement whose truth we can seek establish, then it doesn't need to be said. You were supposed to show all your work in high school algebra, but not in proof-writing.

  • You demonstrate that you understand the central idea of the proof, but it's a bit messy. You'll get better at that over time.

  • I'm a believer that, like closing a set of parentheses or quotation marks, you should be clear when you are done with the scope of your universally quantified variables. The point is that this was true for all $x,y\in A$, so it's worth taking half a sentence to note that you did that any never assumed anything else about $x$ or $y$.

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