Homework 2
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.
Q1. Exercise 6, p. 53, item a)
a)
∃
xN(x)
There exists a student in my school that has visited North Dakota.
Q2. Exercise 6, p. 53, item e)
e) ¬
∀
xN(x)
Not one student in my school has visited North Dakota.
Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the
statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and
logical connectives. Let the domain consist of all students in your class.
Q3. Exercise 10, p. 53. Item a)
A student in my class has a cat, a dog, and a ferret.
∃
x( C(x)
∧
D(x)
∧
F(x) )
Q4. Exercise 10, p. 53. Item c)
Some student in my class has a cat and a ferret, but not a dog.
∃
x( C(x)
∧
F(x)
∧
¬D(x) )
Suppose that the domain of the propositional function P(x) consists of the integers−2,−1, 0, 1, and 2.
Write out each of these propositions using disjunctions, conjunctions, and negations.
Q5. Exercise 18, p. 53, item a)
∃
xP(x)
P(-2)
∨ P(-1) ∨ P(0) ∨ P(1) ∨ P(2)
Q6. Exercise 18, p. 53, item f)
¬
∀
xP(x)
¬P(-2)
∧
¬P(-1)
∧
¬P(0)
∧
¬P(1)
∧
¬P(2)
Suppose that the domain of the propositional function P(x) consists of−5,−3,−1, 1, 3, and 5. Express
these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.