Predicate Calculus- Exercises - Computer Science - University of ...

Predicate Calculus- Exercises
Lila Kari
The University of Western Ontario
Predicate Calculus- Exercises CS2209, Applied Logic for Computer Science 1 / 11

Exercises
1. Translate the following propositions into formulas of L pred (select
suitable symbols). The universe of discourse is the set of all complex
numbers:
• All rational numbers are real numbers.
• Not all real numbers are rational numbers.
• Some real numbers are not rational numbers.
• Every natural number is either odd or even.
• No natural number is both odd and even.
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Exercises from Lewis Carroll
Lewis Carroll, (really Charles Dodgson writing under a pseudonym),
the author of Alice in Wonderland is also author of several works in
symbolic logic. The next exercises come from his book Symbolic
Logic.
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Exercise 1
Consider the following statement. argument.
All lions are fierce.
Some lions do not drink coffee.
Some fierce creatures do not drink coffee.
Express the argument in the language of predicate calculus, where
P(x), Q(x) and R(x) are the statements “x is a lion”, “x is fierce”
and “x drinks coffee” respectively. The universe of discourse is the
set of all creatures.
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Exercise from Lewis Carroll
Let P(x), Q(x) and R(x) be the statements “x is a professor”, “x is
ignorant”, and “x is vain”. Express each of the following statements
in L pred where the universe of discourse is the set of all people.
a) No professors are ignorant.
b) All ignorant people are vain.
c) No professors are vain.
Does c) follow from a) and b)
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Exercise from Lewis Caroll
Let P(x), Q(x), R(x) and S(x) be the statements “x is a baby”, “x
is logical”, “x is able to manage crocodiles” and “x is despised”.
Express each of the following statements in L pred if the universe of
discourse is the set of all people:
a) Babies are illogical.
b) Nobody is despised who can manage a crocodile.
c) Illogical persons are despised.
d) Babies cannot manage crocodiles.
Does d) follow from a), b) and c)
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Exercises
Let P(x) be the statement “x = x 2 ”. If the universe of discourse is
the set of integers, what are the truth values of the following
(a) P(0)
(b) P(1)
(c) P(2).
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Let Q(x, y) be the statement “x + y = x − y”. If the universe of
discourse for both variables is the set of integers, what are the truth
values of the following
a) Q(1, 1) b) Q(2, 0)
c) ∀yQ(1, y) d) ∃xQ(x, 2)
e) ∃x∃yQ(x, y) f ) ∀x∃yQ(x, y)
g) ∃y∀xQ(x, y) h) ∀y∃xQ(x, y)
i) ∀x∀yQ(x, y)
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Exercises
Let L(x, y) be the predicate x loves y, where the domain for both x
and y is the set of all people in the world. Express the following in
predicate calculus.
a) Everybody loves Jerry.
b) Everybody loves somebody.
c) There is somebody whom everybody loves.
d) Nobody loves everybody.
e) There is somebody whom Lydia does not love.
f) There is somebody whom no one loves.
g) There is exactly one person whom everybody loves.
h) There are exactly two people whom Lynn loves.
i) Everyone loves himself or herself.
j) There is somebody who loves no one besides himself or herself.
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Exercises
Express the negation of the following propositions using quantifiers.
Also, express these negations in English. The universe of discourse is
the set of students in this class.
a) Every student in this class likes mathematics.
b) There is a student in this class who has never seen a computer.
c) There is a student in this class who has taken every mathematics
course offered at this school.
d) There is a student in this class who has been in at least one room
of every building on campus.
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Exercises
Express the following statements in predicate calculus. State the
universe of discourse.
a) Every computer science student needs a course in discrete
mathematics.
b) There is a student in this class who owns a personal computer.
c) Every student in this class has taken at least one computer science
course.
d) There is a student in this class who has taken at least one course
in computer science.
e) Every student in this class has been in every building on campus.
f) There is a student in this class who has been in every room of at
least one building on campus.
g) Every student in this class has been in at least one room of every
building on campus.
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