COT 3100 Homework 3

Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement "∀x(R(x) → H(x))" into English.

- Every animal is a rabbit and hops. - Every rabbit hops. - If an animal hops, then it is a rabbit.

Identify the correct steps involved in proving p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent. (Check all that apply.)

- The first statement p ↔ q is true if and only if p and q have the same truth value. - The first statement p ↔ q is true if p and q have different truth values. - If both p and q are false, then (p ∧ q) is false and (¬p ∧ ¬q) is true. This again implies that the second statement (p ∧ q) ∨ (¬p ∧ ¬q) is true. - If both p and q are false, then (p ∧ q) is false and (¬p ∧ ¬q) is true. This again implies that the second statement (p ∧ q) ∨ (¬p ∧ ¬q) is false. - If p is true and q is false, then (p ∧ q) is false and (¬p ∧ ¬q) is true. This again implies that the second statement (p ∧ q) ∨ (¬p ∧ ¬q) is false. - Thus, p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) have same truth value; hence, they are logically equivalent.

contradiction

All false

tautology

All true

contingency

At least one true and at least one false

Translate in two ways the given statement into logical expressions using predicates, quantifiers, and logical connectives. Let C(x) be the propositional function "x is in your class" and P(x) be "x has a cellular phone." Click and drag an expression into each domain so that the expression, subject to the domain, has the meaning "Everyone in your class has a cellular phone." Domain A: The domain consists of the students in your class. Domain B: The domain consists of all people.

Domain A: ∀xP(x) Domain B: ∀x(C(x) → P(x))

Translate these statements into English, where C(x) is "x is a comedian," F(x) is "x is funny," and the domain consists of all people. ∀x(C(x) → F(x))

Every comedian is funny.

Translate these statements into English, where C(x) is "x is a comedian," F(x) is "x is funny," and the domain consists of all people. ∀x(C(x) ∧ F(x))

Every person is a funny comedian.

Find the truth values of these statements if the domain consists of all integers. ∃!x(x > 1)

False

Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(true)?

False

The dual of the compound proposition p ∧ (q ∨ (r ∧ T)) is p ∨ (q ∧ (r ∨ T)).

False

Use De Morgan's laws to find the negation of the given statement. The negation of the statement "Kwame will take a job in industry or go to graduate school" using De Morgan's law is "Kwame will not take a job in industry or will not go to graduate school."

False

Translate these specifications into English where F(p) is "Printer p is out of service," B(p) is "Printer p is busy," L(j) is "Print job j is lost," and Q(j) is "Print job j is queued." ∃p(F(p) ∧ B(p)) → ∃jL(j)

If there is a printer that is both out of service and busy, then some job has been lost.

Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∃xP(x) without using quantifiers, instead use only negations and disjunctions. Click on the choice or choices that correspond to a correct solution.

P(-5) ∨ P(-3) ∨ P(-1) ∨ P(1) ∨ P(3) ∨ P(5)

Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∀xP(x) without using quantifiers, instead use only negations, disjunctions, or conjunctions. Click on the choice or choices that correspond to a correct solution.

P(1) ∧ P(3) ∧ P(5) ∧ P(-5) ∧ P(-3) ∧ P(-1)

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. F(x) be the expression "x has fleas," and the domain of discourse is dogs. The statement is "All dogs have fleas."

The expression is ∀ x F(x), its negation is ∃ x¬F(x), and the sentence is "There is a dog that does not have fleas."

For each of these statements, find a domain for which the statement is true or false. The statement "Everyone is older than 21 years" is _______ in the domain "All United States senators."

True

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." Let the domain consist of all students in your class. The quantification for the below statement is expressed in terms of C(x), D(x), F(x), quantifiers, and logical connectives as ∀x(C(x) ∨ D(x) ∨ F(x)). All students in your class have a cat, a dog, or a ferret.

True

Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(orange)?

True

Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(Denver, Colorado).

True

Use De Morgan's laws to find the negation of the given statement. The negation of the statement "James is young and strong" using De Morgan's law is "James is not young, or he is not strong." Is it true?

Yes

Use De Morgan's laws to find the negation of the given statement. Yoshiko knows Java and calculus.

Yoshiko does not know Java or does not know calculus.

Prove the given expression is a tautology by developing a series of logical equivalence to demonstrate that it is logically equivalent to T. [¬p Λ (p V q)] → q

[¬p Λ (p V q)] → q [(¬p Λ p) V (¬p Λ q)] → q by distributive law [F V (¬p Λ q)] → q by negation law (¬p Λ q) → q by identity law ¬(¬p Λ q) V q (p V ¬q) V q by De Morgan's Law p V (¬q V q) by associative law p V T by negation law T by domination law

Fill the given truth table. p | (p ∧ F) T | a F | b

a: T b: F

Fill the given truth table. p | (p ∧ ⊤) T | a F | b

a: T b: F

Identify the compound proposition involving the propositional variables p, q, and r that is true when p and q are true and r is false, but is false otherwise.

p ∧ q ∧ ¬r

Find the dual of the given compound proposition. p ∨ ¬q

p ∧ ¬q

The compound proposition (p ∨ q ∨ ¬r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (p ∨ ¬r ∨ ¬s) ∧ (¬p ∨ ¬q ∨ ¬s) ∧ (p ∨ q ∨ ¬s) is

satisfiable

Let P(x) be the statement "x > 1." Suppose we initially assign "x := 0" and then execute the statement "if P(x) then x := 1." What is the value of x after the statement is executed?

x = 0

Find a counterexample, to the given universally quantified statements, where the domain consists of all real numbers. ∀x(x2 ≠ x)

x = 1 x = -1 x = - 2-√

Click and drag the steps in the correct order to show that ¬p → (q → r) and q → (p ∨ r) are logically equivalent. (Note: While proving, prove the equivalence from ¬p → (q → r) to q → (p ∨ r).)

¬p → (q → r) ≡ p V (q → r) ≡ p V ¬q V r ≡ ¬q V p V r ≡ q → (p V r)

Consider the following propositions. Let S(x) denote "x obeys the speed limit," where the domain is the set of all drivers. Express the negation of the proposition "Some drivers do not obey the speed limit" using quantifiers, and then express the negation in English.

∀xS(x); All drivers obey the speed limit.

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." Let the domain consist of all students in your class. Express the statement below in terms of C(x), D(x), F(x), quantifiers, and logical connectives. A student in your class has a cat, a dog, and a ferret.

∃x(C(x) ∧ D(x) ∧ F(x))

Consider the set of all propositions. If T(x) means that x is a tautology and C(x) means that x is a contradiction, express each of these statements using logical operators, predicates, and quantifiers. Some propositions are tautologies.

∃xT(x)

Consider the following propositions. Let S(x) denote "x is serious," where the domain is the set of all Swedish movies. Express the negation of the proposition "All Swedish movies are serious" using quantifiers, and then express the negation in English.

∃x¬S(x); Some Swedish movies are not serious.