My apologies beforehand if the question is worded poorly because I've been stumped on the exact meaning of this question for awhile. This was the best way I could phrase it.
I've been stuck in a predicate logic question written as such:
Come up with a model S for which domain is 0, 1.
∀y (∀x.S(y, x) -> ¬∃x.S(y, x))
To my interpretation, the common English wording is:
Come up with a set S for which, for any y, any x implies there does not exist an x.
Now it seems to me that the only set that can always satisfy this condition seems to be the empty set {}.
Rough ideas here:
Initially I have rewritten it to be:
∀y (¬∀x.S(y, x) v ¬∃x.S(y, x))
And my initial thought is:
S = {(0,0),(1,1)}
However, it does not seem to satisfy the first condition and second condition. If y = 0 and x = 0, then clearly there exists an x that matches y.
The empty set seems like the only set that works. However, would there more suitable sets for this predicate logic question?
Sorry I misread the question to be $((\forall y ..) \rightarrow ..)$ instead of $\forall y ( .. \rightarrow ..)$. In this case, the question says,
Pick any $y \in \{ 0 , 1 \}$. Then: if for some $x \in \{ 0, 1 \} \;, (y,x) \in S \Rightarrow (y, \bar{x} ) \notin S$.
So yeah $\{(0,0) \}$ works and so do many other sets.