Question

a. express the quantified statement in an equivalent way, that is, in a way that has exactly the same meaning.
b. write the negation of the quantified statement. (the negation should begin with \all,\ \some,\ or
o.\)
some cars are red.
a. which of the following expresses the quantified statement in an equivalent way?
○ a. there are no cars that are not red.
○ b. at least one car is red.
○ c. not all cars are red.
○ d. no cars are not red.

Explanation:

• For part a:
• The statement "Some cars are red" means that there exists at least one car that is red.
• Option A "There are no cars that are not red" means all cars are red, which is not equivalent.
• Option B "At least one car is red" is equivalent to "Some cars are red" as "some" can be thought of as "at least one".
• Option C "Not all cars are red" means there is at least one car that is not red, which is not equivalent.
• Option D "No cars are not red" means all cars are red, which is not equivalent.
• For part b:
• The negation of "Some cars are red" (where "some" is an existential quantifier) is a universal quantifier statement. Using the rule that the negation of $\exists xP(x)$ (where $P(x)$ is “$x$ is a red car”) is $\forall x

eg P(x)$. In words, “No cars are red” (or “All cars are not red”) is the negation.

Answer:

a. B. At least one car is red.
b. No cars are red.