How to formally state and prove vacuous truth?
How to show in a proof that a statement is vacuously true because "if $\alpha$ then $\beta$", and also prove $\alpha$ is false, in a formal way? and also particularly, how to structure such proofs?
For example to vacuously prove $\forall A. \varnothing \subseteq A$.
Speaking in natural language (I'm new to mathematics), what I want to formally say is:
$\forall x.$ if $x \in \varnothing $ then $x \in A$.
$x \notin \varnothing $: etc. etc.
"if $x \in \varnothing $" is true because the condition is false, because $x \notin \varnothing $, therefore $x \in A$, vacuously.
2 Answers
$\begingroup$Remember: when you have a sentence of the form $$\forall x \in \emptyset \ \ \ \ p(x)$$ this is always true. Some examples: $$\begin{matrix} \forall x \in \emptyset \ \ \ \ x \notin \emptyset \\ \forall x \in \emptyset \ \ \ \ \exists y \in \emptyset : y >x \\ \forall x \in \emptyset \ \ \ \ x \mbox{ is a beautiful horse} \\ \forall x \in \emptyset \ \ \ \ x=1 \\ \forall x \in \emptyset \ \ \ \ \mbox{ my mother told me to go to school} \\ \forall x \in \emptyset \ \ \ \ \mbox{ Riemann's hypothesis is false} \end{matrix}$$ are all true sentences.
This property allows you to prove some nice properties of the empty set. For example:
$\forall x \in \emptyset \ \ x \in A$, means that $\emptyset \subseteq A$, whenever $A$ is a set.
The empty set is an open set of $\mathbb{R}$. In fact it is true that $\forall x \in \emptyset \ \ \exists r >0 : B(x,r) \subseteq \emptyset$
The empty set is totally ordered. In fact $\forall x,y \in \emptyset \ \ x \leq y$
The supremum of the empty set is $- \infty$. In fact $\forall x \in \emptyset \ \ \forall y \in \mathbb{R} \ \ x \leq y$, so the set $\{ y \in \mathbb{R}: \forall x \in \emptyset \ \ x \leq y \}$ is $\mathbb{R}$, and its "minimum" is $- \infty$
"Vacuous Truth" refers to members of the empty set having some sort of characteristic; normally we do not talk about this in mathematics (it's somewhat useless, really).
For example, $P(x)$ could mean "All the beer in an Alcoholics Anonymous meeting" and $Q(x)$ could mean "is brewed in Mexico." The statement "All the beer in an AA meeting is brewed in Mexico" would be true, despite the fact that no one at an AA meeting drinks beer, and therefore, there's no beer at all in the meeting.
Written in mathematical/logical terms, I believe it would look something like this: $$P(x) \implies Q(x), \space \space\forall x: \lnot P(x)$$
Another intuitive way to express it:
Let $P$ be the set of all beer at an AA meeting. (Hopefully) $P$ would be the empty set. $Q(x)$ denotes the beer $x$ brewed in Mexico. Thus, our statement would be expressed as: $$\forall x \in P, \space \space Q(x), $$ where $P = \emptyset$.
I think you're more likely to see this logical structure in more classical/theoretical works in logic rather than mathematics. As far as my knowledge is concerned, the "null set" is a bit of a dead end: mathematicians don't talk about "elements in a null set" for the sake of pointing out "vacuous truth."
If you'd like more information, this site should be a good reference:
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