مخظنالاتار
P1: 1/1
CH01-7T
P2: 1/2
QC: 1/1
Rosen-2311T
T1: 2
MHIA017-Rosen-v5.cls
May 13, 2011
15:27
1 - 2 (c,d) - 5 (c) - 7 (b) - 11 - 17 (a,f) - 23 (d,e) - 33 1.4 Predicates and Quantifiers
(c) 41 (a) - 61 (d)
Exercises
1. Let P (x) denote the statement “x ≤ 4.” What are these truth values?
a) P (0)
b) P (4)
c) P (6)
2. Let P (x) be the statement “the word x contains the letter a.” What are these truth values?
a) P (orange)
b) P (lemon)
c) P (true)
d) P (false)
3. Let Q(x, y) denote the statement “x is the capital of y.”
What are these truth values?
a) Q(Denver, Colorado)
b) Q(Detroit, Michigan)
c) Q(Massachusetts, Boston)
d) Q(New York, New York)
4. State the value of x after the statement if P (x) then x := 1 is executed, where P (x) is the statement “x > 1,” if the value of x when this statement is reached is
a) x = 0.
b) x = 1.
c) x = 2.
5. Let P (x) be the statement “x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in
English.
a) ∃xP (x)
b) ∀xP (x)
c) ∃x ¬P (x)
d) ∀x ¬P (x)
6. Let N(x) be the statement “x has visited North Dakota,” where the domain consists of the students in your school.
Express each of these quantifications in English.
a) ∃xN(x)
b) ∀xN (x)
c) ¬∃xN (x)
d) ∃x¬N(x)
e) ¬∀xN (x) f ) ∀x¬N(x)
7. Translate these statements into English, where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.
a) ∀x(C(x) → F (x))
b) ∀x(C(x) ∧ F (x))
c) ∃x(C(x) → F (x))
d) ∃x(C(x) ∧ F (x))
8. Translate these statements into English, where R(x) is “x is a rabbit” and H (x) is “x hops” and the domain consists of all animals.
a) ∀x(R(x) → H (x))
b) ∀x(R(x) ∧ H (x))
c) ∃x(R(x) → H (x))
d) ∃x(R(x) ∧ H (x))
9. 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),
CH01-7T
P2: 1/2
QC: 1/1
Rosen-2311T
T1: 2
MHIA017-Rosen-v5.cls
May 13, 2011
15:27
1 - 2 (c,d) - 5 (c) - 7 (b) - 11 - 17 (a,f) - 23 (d,e) - 33 1.4 Predicates and Quantifiers
(c) 41 (a) - 61 (d)
Exercises
1. Let P (x) denote the statement “x ≤ 4.” What are these truth values?
a) P (0)
b) P (4)
c) P (6)
2. Let P (x) be the statement “the word x contains the letter a.” What are these truth values?
a) P (orange)
b) P (lemon)
c) P (true)
d) P (false)
3. Let Q(x, y) denote the statement “x is the capital of y.”
What are these truth values?
a) Q(Denver, Colorado)
b) Q(Detroit, Michigan)
c) Q(Massachusetts, Boston)
d) Q(New York, New York)
4. State the value of x after the statement if P (x) then x := 1 is executed, where P (x) is the statement “x > 1,” if the value of x when this statement is reached is
a) x = 0.
b) x = 1.
c) x = 2.
5. Let P (x) be the statement “x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in
English.
a) ∃xP (x)
b) ∀xP (x)
c) ∃x ¬P (x)
d) ∀x ¬P (x)
6. Let N(x) be the statement “x has visited North Dakota,” where the domain consists of the students in your school.
Express each of these quantifications in English.
a) ∃xN(x)
b) ∀xN (x)
c) ¬∃xN (x)
d) ∃x¬N(x)
e) ¬∀xN (x) f ) ∀x¬N(x)
7. Translate these statements into English, where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.
a) ∀x(C(x) → F (x))
b) ∀x(C(x) ∧ F (x))
c) ∃x(C(x) → F (x))
d) ∃x(C(x) ∧ F (x))
8. Translate these statements into English, where R(x) is “x is a rabbit” and H (x) is “x hops” and the domain consists of all animals.
a) ∀x(R(x) → H (x))
b) ∀x(R(x) ∧ H (x))
c) ∃x(R(x) → H (x))
d) ∃x(R(x) ∧ H (x))
9. 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),