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 computers are smart.
a. which of the following expresses the quantified statement in an equivalent way?
○ a. all computers are not smart.
○ b. at least one computer is smart.
○ c. there are no computers that are not smart.
○ d. not all computers are smart.
b. which of the following is the negation of the quantified statement?
○ a. no computers are smart.
○ b. some computers are not smart.
○ c. all computers are not smart.
○ d. some computers are smart.

Explanation:

• Part a:
• The statement "All computers are smart" means there is no computer that is not smart.
• Option A ("All computers are not smart") is the opposite of the original statement.
• Option B ("At least one computer is smart") is a weaker statement than "All computers are smart".
• Option D ("Not all computers are smart") is also the opposite of the original statement.
• Part b:
• The negation of "All computers are smart" is that there exists at least one computer that is not smart, which is what "Some computers are not smart" means.
• Option A ("No computers are smart") is too extreme.
• Option C ("All computers are not smart") is a universal negative, not the correct negation of a universal affirmative.
• Option D ("Some computers are smart") is not the negation.

Answer:

a. C. There are no computers that are not smart.
b. B. Some computers are not smart.