Express each of these statements using quantifiers. Then form the negation of th
ID: 3143962 • Letter: E
Question
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”)
a) No one has lost more than one thousand dollars playing the lottery.
b) There is a student in this class who has chatted with exactly one other student.
c) No student in this class has sent e-mail to exactly two other students in this class.
d) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book.
Explanation / Answer
(a). Let P(x, y) = a person x has lost y dollars playing the lottery.
then given statement ¬xy(y > 1000 P(x, y)).
negation xy(y > 1000 P(x, y))
in words someone has lost more than $1000 playing the lottery.
(b). Let Q(x, y) = person x has chatted with person y.
our statement xy(y 6= x z(z 6= x (z = y Q(x, z)))).
negation xy(y 6= x z(z 6= x ¬(z = y Q(x, z)))).
in words everybody in this class has either chatted with no one else or has chatted with two or more others.
(c). Let R(x, y)= person x has sent e-mail to person y.
statement ¬xyz(y 6= z x 6= y x 6= z w(w 6= x (R(x, w) (w = y w = z)))).
negation xyz(y 6= z x 6= y x 6= z w(w 6= x (R(x, w) (w = y w = z)))).
In words some student in this class has sent e-mail to exactly two other students in this class.
(d). Let S(x, y)= student x has solved exercise y.
statement xyS(x, y).
negation xy¬S(x, y).
In words for every student in this class, there is some exercise that he or she has not solved.
(e). Let S(x, y)= mean that student x has solved exercise y,
and let B(y, z)= exercise y is in section z of the book.
statement ¬xzy(B(y, z) S(x, y)).
negation xzy(B(y, z) S(x, y)).
in words some student has solved at least one exercise in every section of this book.
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