Give an example of a predicate P, a nonempty set S, and an expression ϕ containing x as a free variable such that the proposition from Exercise 3.136 is false. Because x has to get its meaning from...

Give an example of a predicate P, a nonempty set S, and an expression ϕ containing x as a free variable such that the proposition from Exercise 3.136 is false. Because x has to get its meaning from somewhere, we will imagine a universal quantifier for x wrapped around the entire expression. Specifically, give an example of P, ϕ, and S for which

Exercise 3.136

Explain why the following propositions are theorems of predicate logic, assuming that x does not appear as a free variable in the expression ϕ (and assuming that S is nonempty):