Exercise Week 2

Question 1

1. Suppose that the domain of the propositional function P (x) consists of the integers 0, 1, 2, 3, and 4.
   Write out each of these propositions using disjunctions, conjunctions, and negations.

Answer 1

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Question 2

1a(Thursday). ∃x P(x)

Answer 2

P(0) ∨…. ∨P(4)

Question 3

1b(Thursday) .∀x P(x)

Answer 3

P(0)∧…∧P(4)

Question 4

2. Let P (x) be the statement “x can speak Russian” and let Q(x) be the statement “x knows the computer language C++.” Express each of these sentences in terms of P (x), Q(x), quantifiers, and logical connectives. The domain for quantifiers consists of all students at your school.

Answer 4

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Question 5

2a (Thursday). There is a student at your school who can speak Russian and who knows C++.

Answer 5

∃x (P(x) ∧ Q(x))

Question 6

1c. ∃x ¬ P(x)

Answer 6

¬P(0) ∨… ∨¬P(4)

Question 7

2b (Thursday). There is a student at your school who can speak Russian but who doesn’t know C++.

Answer 7

∃x (P(x) ∧ ¬Q(x))

Question 8

2c (Thursday). Every student at your school can speak Russian or knows C++.

Answer 8

∀x (P(x) ∨ Q(x))

Question 9

2d (Thursday). No student at your school can speak Russian or knows C++.

Answer 9

¬∃x (P(x) ∨ Q(x)) == ∀x (¬P(x) ∧ ¬Q(x))

Question 10

3.Translate in two ways each of these statements into logical expressions using predicates, quantifiers,
and logical connectives. First, let the domain consist of the students in your class and second, let it
consist of all people.

Answer 10

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Question 11

3a (Thursday). Someone in your class can speak Hindi

Answer 11

1. ∃x(H(x))
2. ∃x(C(x) ∧ H(x))

Question 12

3b (Thursday). Everyone in your class is friendly

Answer 12

1. ∀x (F(x))
2. ∀x (C(x) → F(x))

Question 13

3c (Thursday). There is a person in your class who was not born in California

Answer 13

1.∃x ¬B(x)
2.∃x(C(x) ∧ ¬B(x))

Question 14

3d (Thursday). A student in your class has been in a movie

Answer 14

1. ∃x M(x)
2. ∃x (C(x) ^ M(x))

Question 15

3e (Thursday). No student in your class has taken a course in logic programming

Answer 15

1. ¬∃x P(x) or ∀x ¬P(x)
2. ¬∃x (C(x) ^ P(x)) or ∀x (C(x) → ¬P(x))

Question 16

4. Translate each of these statements into logical expressions using predicates, quantifiers, and logical
   connectives.

Answer 16

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Question 17

4a (Thursday). No one is perfect

Answer 17

¬ ∃x P(x) = ∀x ¬ P(x)

Question 18

4b (Thursday). Not everyone is perfect

Answer 18

c ∀x P(x) = ∃x ¬ P(x)

Question 19

4c (Thursday). All your friends are perfect.

Answer 19

∀x (F(x) → P(x))

Question 20

4d (Thursday). At least one of your friends is perfect.

Answer 20

∃x (F(x) ^ P(x))

Question 21

4e (Thursday). Everyone is your friend and is perfect

Answer 21

∀x (F(x) ^ P(x))

Question 22

4f (Thursday). Not everybody is your friend or someone is
not perfect.

Answer 22

¬ ∀x F(x) ∨ ∃x ¬ P(x)

Question 23

5. 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.”) Domain: All Creatures

Answer 23

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Question 24

5a. Some old dogs can learn new tricks.

Answer 24

1. ∃x (D(x) ^ T(x))
2. ∀x ( D(x) → ¬ T(x))
3. For all creatures. if you are an old dog you cannot learn new tricks.

Question 25

5b. No rabbit knows calculus.

Answer 25

1. ¬ ∃x (R(x) ^ C(x))
   2.∃x (R(x) ^ C(x))

Question 26

5c. Every bird can fly.

Answer 26

1. ∀x (B(x) → F(x))
2. ∃x( B(x) ^ ¬ F(x))

Question 27

5d. There is no dog that can talk.

Answer 27

1. ¬ ∃x ( D(x) ^ T(x))
2. ∃x ( D(x) ^ T(x))

Question 28

5e. There is no one in this class who knows French and Russian

Answer 28

1. ¬∃x(C(x) ^ (F(x) ^ R(x)))
2. ∃x(C(x) ^ (F(x) ^ R(x)))

Question 29

6. Express the negation of each of these statements in terms of quantifiers without using the negation
   symbol.

Answer 29

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Question 30

6a.¬∀x(x > 1)

Answer 30

∃x( x <= 1)

Question 31

6b.

Answer 31

∃x( x > 1)

Question 32

6c.

Answer 32

∀x( x < 4)

Question 33

6d.

Answer 33

∀x( x >= 0)

Question 34

6e.

Answer 34

∃x((x >= 1) ^ (x <= 2))

Question 35

6f

Answer 35

∀x(( x >= 4) ^ (x <= 7))

Question 36

Bonus: ¬∀x(1 <= x < 2)

Answer 36

∃x ¬( (x >= 1) ^ (x < 2))