I'm revising for an upcoming maths assessment for a theory of algorithms module, but I'm not entirely sure if my solutions to the below questions are correct.
My main confusion with the questions is using this S(x,y) statement.
Can anyone review my below answers and correct any errors if present?
These are my solutions to the above questions:
(i) ¬∃ x ∈ C [F(x) ^ P(x)]
(ii)∃ x ∈ C [ P(x)[S(x,y)]]
(iii)∀ F(x) ∈ C [ P(x)[S(x,y)]]
(iv)∀ F(x) ∈ C [ P(x)¬[S(x,y)]]
(i) $¬∃ x ∈ C \;[F(x) \wedge P(x)]$
$\color{green}{\checkmark}$ "No car is both Ferrari and Porsche."
(ii) $∃ x ∈ C [ P(x)[S(x,y)]]$
$\color{red}{\times}$ "some car is a Porsche 'times?' slower than free y"
You need to use a conjunction between the predicates rather than ...multiplying?..., and you must quantify the free variable (and restrict it to Ferrari). Thus:
$$\exists x \in C\; \exists y\in C \; [P(x)\wedge F(y)\wedge S(x,y)]$$
(iii) $∀ F(x) ∈ C [ P(x)[S(x,y)]]$
$\color{red}{\times}$ This doesn't even parse. "for all ($x$ is a Ferrari) in cars then $x$ is a Porsche 'times?' $x$ is slower than $y$" ?
(iv) $∀ F(x) ∈ C\; [ P(x)¬[S(x,y)]]$
$\color{red}{\times}$ "for all ($x$ is a Ferrari) in cars then $x$ is a Porsche $x$ 'times?' $x$ is not slower than $y$"
A predicate does not belong in the quantification clause, you need to quantify both entities, $x$ and $y$, and to use logical connectives rather than ...multiplying?... the predicates.
Remember too that the connective used for a restricted existential is conjunction ($\exists z\; [R(z)\wedge Q(z) ]$) while the connective used for a restricted universal is implication ($\forall z\; [R(z) \to Q(z)]$)