- Let $x_1,\ldots,x_5$ be five variables. Write a formula using only these variables and the operators $\lor$, $\land$, and $\lnot$ equivalent to the sentence "At least two and at most three of these variables are true".
The first thing that came to my mind is that this sentence is self-referential, so it shouldn't be considered a proposition. But I'm probably missing something here.
- Translate "A day next month is Sunday" defining the predicates and domain of quantifiers.
Let $S(x)$ be "$x$ is Sunday". A silly first approach was to define a "convenient" universe of discourse: let $D = \{d : d \text{ is day of the next month}\}$, then the predicate was $\exists x \in D(S(x))$. But "next month" seemed somewhat open, so this bothered me. Another try was: let $M = \{m : m \text{ is a month of the year}\}$ and $D_m = \{d : \exists m \in M(d \text{ is day of } m)\}$. Thus I could write $\exists x \in D_{m+1}(S(x))$. But I have the sensation that this notation overcomplicates the solution. What do you think? Is there a cleaner way to express this?