Question

a. express the following quantified statement in an equivalent way, that is, in a way that has exactly the same meaning.
b. write the negation of the following quantified statement. (the negation should begin with \all,\ \some, \ or
o.\)
not all people are breathing.
a. express the quantified statement in an equivalent way.
all people are not breathing.
at least one person is breathing.
there are no people that are not breathing.
some people are not breathing.
b. write the negation of the quantified statement.
all people are not breathing.
all people are breathing.
some people are breathing.
some people are not breathing.

Explanation:

• For part a:
• The statement "Not all people are breathing" means that there exists at least one person who is not breathing. In logical terms, "Not all \(x\) are \(P(x)\)" is equivalent to "Some \(x\) are not \(P(x)\)".
• For part b:
• The negation of "Not all people are breathing" (which is equivalent to "Some people are not breathing") is "All people are breathing". In logical terms, the negation of "\(\exists x(

eg P(x))\)" (where \(x\) represents people and \(P(x)\) represents "is breathing") is "\(\forall xP(x)\)"

Answer:

a. Some people are not breathing.
b. All people are breathing.