Consider the formula F =∀x ∃y ∃z (¬(x=y) ∧ ¬P(x,y) ∧ ¬(x=z) ∧ P(x,z)).
I have to find a universe $U$ which cardinality is given by n, while n >= 3. Furthermore I should find a predicate $P: U^{2} \rightarrow \{0,1\} $ which satisfies the given formula F. The universe should be defined in terms of n, meaning that if I would tell you n = 5 everything should work. I tried several things but don't understand how to define this universe in terms of n.
I actually now found a solution: Consider $U = \{n-n, n-(n-1), ..., n-1 \;|\; n \in N, \; n \geq 3 \}$ with $$ P(a, b) =\begin{cases}1,\;x + 1 \equiv y \;(mod\; 3)\\0,\; otherwise\end{cases} $$