I have a quick question. I've been studying a little boolean algebra and noticed that the basic operations seem to correlate well with set operations.
So to extrapolate, I tried to visualize True and False as sets. The only way I got this to work is considering True as the universal set and False as the empty set. Is this the right and only way of thinking about it?
On
Let $p(x)$ a predicate. Let's call the set of $x$'s for which $p(x)$ holds true $P$: $$P=\{x|p(x)\}$$ Then trivially: $$p(x)\equiv x\in P$$
Now it follows $$\begin{equation} \begin{split} \neg p(x)&\equiv x\in P^c\\ p(x)\vee q(x)&\equiv x\in P\cup Q\\ p(x)\wedge q(x)&\equiv x\in P\cap Q\\ \end{split} \end{equation}$$ where $q(x)\equiv x\in Q$
You can see that $$\operatorname{True}\equiv p(x)\vee \neg p(x)\equiv x\in P\cup P^c\equiv x\in 1$$ $$\operatorname{False}\equiv p(x)\wedge\neg p(x)\equiv x\in P\cap P^c=x\in\emptyset$$
where $1$ is the universe.
For a set $A$ in a universal set $U$, it may be best to think of each individual element of $U$ being associated with True (if the element is in $A$) and False (if the element is not in $A$).