Reading Quiz Section 2.2
1. Let P(x) be the proposition x
2
− 1 = 0, with domain all real numbers. Which of the following
statements are true?
(a) P(1) (b) P( −1) (c) P(3) (d) P(x)
(e) ∀x, P(x) (e) ∃x, P(x) (f) ¬
∀x, P(x)
2. A value x
0
in the domain of P for which P(x
0
) is false is known as a(n)
(a) example (b) counterexample
(c) realization (d) solution
3. Which of the following are equivalent to the given expression?
¬
∀x, ∃y, P (x, y)
(a) ∃x, ∀y, P(x, y) (b) ¬
∃x, ∀y, P(x, y)
(c) ∃x, ∀y, ¬P(x , y) (d) ∀x, ∃y, ¬P(x , y)
4. True or False: the order of quantifiers in an expression can always be switched without chang-
ing the meaning of the expression.
Practice Problems Section 2.2
1. Write each of the following using propositional functions and quantifiers. Make sure to define
any propositional functions you are using.
(a) Every class has an instructor.
(b) For all real numbers x and y, if x and y are positive, then there exists a positive integer n
such that nx > y.
(c) For each positive integer n, there exists a real number which is positive and is less than
1
n
.
Video Solution
2. Negate each proposition (a), (b), (c) in the previous problem.
Video Solution
3. Which of the following propositions are true and which false? Justify your answers.
(a) ∀x ∈ R, ∃y ∈ R, y
4
= 4x
(b) ∃y ∈ R, ∀x ∈ R, y
4
= 4x
(c) ∀y ∈ R, ∃x ∈ R, y
4
= 4x
(d) ∃x ∈ R, ∀y ∈ R, y
4
= 4x
Video Solution
