Question

Express each of these statements using quantifiers. Then form the negation of the statement sothat no negation is to the left of a quantifier. Next, express the negation in simple English. In eachcase, identify the domain and specify the predicates.

•No one has lost more than one thousand dollars playing the lottery.

•There is a student in this class who has chatted with exactly one other student.

•No student in this class has sent e-mail to exactly two other students in this class.

• Some student has solved every exercise in this book.

•No student has solved at least one exercise in every section of this book.

Answer 1

Solution

No one has lost more than one thousand dollars playing the lottery.

Let L(x, y) mean that person x has lost y dollars playing the lottery.

The original statement is then ¬∃x∃y(y > 1000 ∧ L(x, y)).

Its negation of course is ∃x∃y(y > 1000 ∧ L(x, y))

someone has lost more than 1000 dollars playing the lottery.

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There is a student in this class who has chatted with exactly one other student.

Let C(x, y) mean that person x has chatted with person y.

The given statement is ∃x∃y(y ¹ x ∧ ∀z(z ¹ x → (z = y ↔ C(x, z)))).

The negation is therefore ∀x∀y(y ¹ x → ∃z(z ¹ x ∧ ¬(z = y ↔ C(x, z)))).

In English, everybody in this class has either chatted with no one

else or has chatted with two or more others.

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No student in this class has sent e-mail to exactly two other students in this class

Let E(x, y) mean that person x has sent e-mail to person y.

The given statement is ¬∃x∃y∃z(y ¹ z ∧ x ¹ y ∧ x ¹ z ∧ ∀w(w ¹ x → (E(x, w) ↔ (w = y ∨ w = z)))).

The negation is obviously

∃x∃y∃z(y ¹ z ∧ x ¹ y ∧ x ¹ z ∧ ∀w(w ¹ x → (E(x, w) ↔ (w = y ∨ w = z)))).

In English,

some student in this class has sent e-mail to exactly two other students in this class.

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Some student has solved every exercise in this book.

Let S(x, y) mean that student x has solved exercise y.

The statement is ∃x∀yS(x, y).

The negation is ∀x∃y¬S(x, y).

In English, for every student in this class, there is some exercise that he or she has not solved.

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No student has solved at least one exercise in every section of this book.

Let S(x, y) mean that student x has solved exercise y, and let B(y, z) mean that exercise y

is in section z of the book.

The statement is ¬∃x∀z∃y(B(y, z) ∧ S(x, y)).

The negation is of course ∃x∀z∃y(B(y, z) ∧ S(x, y)).

In English, some student has solved at least one exercise in every section of this book.

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all the best