# Predicate logic and quantifiers: solutions to the textbook exercises

In this article, you will see a video explaining predicate logic and quantifiers. After that, I’ll show the solution to the first five exercises from the text-book that professors are using to teach Discrete Mathematics in most universities.

Find below the solution to the first five exercises from the text-book: Discrete Mathematics and Application by Rosen.

## 1- Let P(x) denote the statement “x ≤ 4.” What are these truth values?

a) P(0)

b) P(4)

c) P(6)

``````a) P(0) = “0≤ 4”. Therefore, the truth value is true.
b) P(4) = “4≤4”. Therefore, the truth value is true.
c) P(6) = “6≤4”. Therefore, the truth value is false.``````

## 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)

``````a) P(orange) = “the word orange contains the letter a.”. Therefore, the truth value is true.
b) P(lemon) = “the word lemon contains the letter a.”. Therefore, the truth value is false.
c) P(true) = “the word true contains the letter a.”. Therefore, the truth value is false.
d) P(false) = “the word false contains the letter a.”. Therefore, the truth value is true.``````

## 3- Let Q(x, y) denote the statement “x is the capital of y.” What are these truth values?

b) Q(Detroit, Michigan)

c) Q(Massachusetts,Boston)

d) Q(NewYork,NewYork)

``````a) Q(Denver,Colorado) = “Denver is the capital of Colorado.” True.
b) Q(Detroit, Michigan) = “Detroit is the capital of Michigan.” False, the capital of Michigan is Lansing.
c) Q(Massachusetts, Boston) = “Massachusetts is the capital of Boston.” False. Boston is the capital of Massachusetts.
d) Q(NewYork, NewYork) = “NewYork is the capital of NewYork.” True.``````

## 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.

``````a) x=0.  x equals 0 before the condition. The statement “0>1” is false, therefore the value of x does not change.
b) x=1. x equals 1 before the condition. The statement “1>1” is false, therefore the value of x does not change.
c) x=2. x equals 2 before the condition. The statement “2>1” is true, therefore the value of x change to 1.``````

## 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) b) ∀x ¬P(x)

``````a) There is at least one student that spends more than five hours every weekday in class.