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 math tests are not fun.
a. which of the following expresses the quantified statement in an equivalent way?
a. all math tests are not fun.
b. there are no math tests that are not fun.
c. not all math tests are fun.
d. at least one math test is fun.

Explanation:

• For part a:
• The statement "Some math tests are not fun" means that there exists at least one math test that is not fun. "Not all math tests are fun" also implies that there is at least one math test that fails to be fun. "All math tests are not fun" is a stronger statement (implying every math test is not fun). "There are no math tests that are not fun" is the opposite of the original statement. "At least one math test is fun" is about a positive attribute (fun) for at least one test, which is not equivalent to the original negative - about - some statement.
• For part b:
• The negation of "Some \(x\) are not \(y\)" (where \(x\) is math tests and \(y\) is fun) is "All \(x\) are \(y\)". Using the rules of quantifier negation (\(

eg\exists x(
eg P(x))\equiv\forall xP(x)\) in predicate logic, where \(P(x)\) is "math test \(x\) is fun").

Answer:

a. C. Not all math tests are fun.
b. All math tests are fun.