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.\)
all beetles are carnivorous.

a. which of the following expresses the quantified statement in an equivalent way?
a. there are no beetles that are not carnivorous.
b. not all beetles are carnivorous.
c. all beetles are not carnivorous.
d. at least one beetle is carnivorous.

Explanation:

• For part a:
• The statement "All beetles are carnivorous" means that there is no beetle that does not have the property of being carnivorous.
• Option A "There are no beetles that are not carnivorous" is equivalent. Option B "Not all beetles are carnivorous" is the negation of the original statement. Option C "All beetles are not carnivorous" is a universal negative which is not equivalent. Option D "At least one beetle is carnivorous" is an existential positive which is not equivalent to the universal positive.
• For part b:
• The negation of a universal positive statement “All \(S\) are \(P\)” (where \(S\) is beetles and \(P\) is carnivorous) is an existential negative statement. Using the rules of quantifier negation \(

eg(\forall x(S(x)\to P(x)))\equiv\exists x(S(x)\land
eg P(x))\). In words, “Some beetles are not carnivorous”

Answer:

a. A. There are no beetles that are not carnivorous.
b. Some beetles are not carnivorous.