is ∀x (¬P (x) ∨ ¬S(x)∨¬F (x)) or ∀x (P (x) → (¬ S(x) ∨ ¬ F (x)).
In English: “Every pig either can’t swim or it can’t catch fish”.
1.5.8 Let Q(x, y) be the statement “student x has been a contestant on quiz
show y”. Express each of these sentences in terms of Q(x, y), quan-
tifiers, and logical connectives, where the domain for x consists of all
students at your school and for y consists of all quiz shows on televi-
sion.
a) There is a student at your school who has been a contestant on a
television quiz show. ∃ x ∃ y Q(x, y).
b) No student at your school has ever been a contestant on a televi-
sion quiz show. ∀ x ∀y ¬ Q(x, y).
c) There is a student at your school who has been a contestant on
Jeopardy and on Wheel of Fortune.
∃x
Q(x, Jeopardy) ∧ Q(x, Wheel of Fortune)
.
d) Every television quiz show has had a student from your school as
a contestant.
∀ y ∃ x Q(x, y).
e) At least two students from your school have been contestants on
Jeopardy.
∃ x ∃z (x 6= z) ∧ Q(x, Jeopardy) ∧ Q(z, Jeopardy).
1.5.10
∗∗
Let F (x, y) be the statement “x can fool y”, where the domain consists
of all people in the world. Use quantifiers to express each of these
statements.
a) Everybody can fool Fred. ∀ x F (x, Fred)
b) Evelyn can fool everybody. ∀ y F (Evelyn, y)
c) Everybody can fool somebody. ∀x ∃ y F (x, y)
d) There is no one who can fool everybody. ¬ ∃ x ∀ y F (x, y)
e) Everyone can be fooled by somebody. ∀ y ∃ x F (x, y)
f) No one can fool both Fred and Jerry. ¬ ∃ x (F (x, Fred) ∧ F (x, Jerry)
g) Nancy can fool exactly two people.
∃ y ∃ z
(y 6= z) ∧ F (Nancy, y) ∧ F (Nancy, z) ∧ ∀ w ((w = y) ∨
(w = z) ∨ ¬ F (Nancy, w))
h) There is exactly one person whom everybody can fool.
∃y
∀x F (x, y) ∧ ( ∀z ((∀wF(w, z)) → y = z)
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