Propositional Logic with Two Statements

Question

I am given two statements. Letting

$s(x)$ denote "$x$ is a car" and $h(x)$ denote "$x$ is manual"

I have to formalise the following statements:

1. "Some car is manual"

Which I think can be denoted as $\exists x\; (s(x) \land h(x))$, but I'm unsure if this is the correct way to put the two statements together.

2. "Not all cars are manual"

3. "All cars are automatic"

4. "There is an automatic car"

5. "Every car is manual"

Answer 1

1."Some car is manual"

Which I think can be denoted as $∃x(s(x)∧h(x))$ , but I'm unsure if this is the correct way to put the two statements together.

$\checkmark$ Yes.   Literally this is "There exists a thing: it is a car and it is manual."   This is indeed "Some car is manual."

2."Not all cars are manual"

What's confused me with that one is whether to use $∃$ or $∀$ . I've thought it could be written as, possibly, $¬∀x(h(x)∧s(x))$ or maybe as $∀¬x(h(x)∧s(x))$

Just do it one bit at a time. "Not"~"$\neg$", "all"~"$\forall x$" , "cars are"~"$s(x)\to$" , "manual"~"$h(x)$". $$\neg\forall x\big(s(x)\to h(x)\big)$$

"Not (every thing: if it is a car, then it is manual)."

Key point: remember that

• "all things in the category have the property" is $\forall x~\big(C(x)\quad\to\quad P(x)\big)$, and
• "some thing in the category has the property" is $\exists x~\big(C(x)\quad\wedge\quad P(x)\big)$

Can you try your hand at the rest?