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 dogs are not rabbits.

a. which of the following expresses the quantified statement in an equivalent way?
○ a. at least one dog is a rabbit.
○ b. there are no dogs that are not rabbits.
○ c. not all dogs are rabbits.
○ d. all dogs are not rabbits.

b. which of the following is the negation of the quantified statement?
○ a. some dogs are rabbits
○ b. not all dogs are rabbits
○ c. all dogs are rabbits
○ d. no dogs are rabbits

Explanation:

Part a

• Analyze Option A: "At least one dog is a rabbit" means some dogs are rabbits, which is different from "Some dogs are not rabbits". Eliminate A.
• Analyze Option B: "There are no dogs that are not rabbits" means all dogs are rabbits, opposite of the original statement. Eliminate B.
• Analyze Option C: "Not all dogs are rabbits" implies that there are some dogs that are not rabbits, which is equivalent to the original statement "Some dogs are not rabbits".
• Analyze Option D: "All dogs are not rabbits" means no dog is a rabbit, which is a stronger statement than "Some dogs are not rabbits". Eliminate D.

The original statement is "Some dogs are not rabbits" (an existential statement with negation in the predicate). The negation of "Some \( x \) are not \( y \)" is "All \( x \) are \( y \)". So we check the options:

• Option A: "Some dogs are rabbits" is not the negation. Eliminate A.
• Option B: "Not all dogs are rabbits" is equivalent to the original statement, not its negation. Eliminate B.
• Option C: "All dogs are rabbits" is the negation of "Some dogs are not rabbits" (since if it's not true that some dogs are not rabbits, then all dogs must be rabbits).
• Option D: (Assuming a typo, likely "No dogs are rabbits" which is not the negation). So the correct negation is "All dogs are rabbits".

Answer:

C. Not all dogs are rabbits.

Part b