EXAM 1

match the universal quantifiers: ∀x (x2 < -1) ∀x (x >0) ∀x (x < 0) ∀x (x2 = 1)

the empty set the positive integers the negative integers {-1,1}

negation of "the summer in Maine is hot and sunny"

the summer in maine is not hot or it is not sunny

Which of the following sentences in propositional logic are contradictions? Multiple select question. p ∨ p ¬p ¬(p → p) ¬p ∧ p

¬(p → p) ¬p ∧ p p ∨ p Reason: This sentence is true when p is true and false when p is false. ¬p Reason: This sentence can be either true or false, depending on the truth value of p. ¬(p → p) Reason: Since p → p is always true, ¬(p → p) is always false. Therefore, it is a contradiction. ¬p ∧ p Reason: If p is true, ¬p must be false. By the truth table for conjunction, the compound expression ¬p ∧ p is false. A similar argument holds for the case where p is false. Therefore, it is a contradiction.

In which of the following expressions is the variable x free? Multiple select question. ∀x P(x) ∨ ∃x (Q(x) → P(x)) ∀y (P(x) ∧ Q(y)) ∃y P(y) ∧ Q(x) ∃x (P(x) ∧ Q(x))

∀y (P(x) ∧ Q(y)) ∃y P(y) ∧ Q(x)

Match each quantified statement having the domain consisting of the integers from 3 to 5 on the left with its corresponding propositional logic expression on the right. Instructions ∃x (x > 4) ∃x(x> 4) drop zone empty. ∃x (4 > x) ∃x(4 >x) drop zone empty. ∀x (x > 4) ∀x(x> 4) drop zone empty. ∀x (4 > x)

∃x (x > 4) matches Choice, (3 > 4) ∨ (4 > 4) ∨ (5 > 4) (3 > 4) ∨ (4 > 4) ∨ (5 > 4) ∃x (4 > x) matches Choice, (4 > 3) ∨ (4 > 4) ∨ (4 > 5) (4 > 3) ∨ (4 > 4) ∨ (4 > 5) ∀x (x > 4) matches Choice, (3 > 4) ∧ (4 > 4) ∧ (5 > 4) (3 > 4) ∧ (4 > 4) ∧ (5 > 4) ∀x (4 > x) matches Choice, (4 > 3) ∧ (4 > 4) ∧ (4 > 5) (4 > 3) ∧ (4 > 4) ∧ (4 > 5)

Which of the following compound propositions are satisfiable? Multiple select question. (¬p ∨ ¬q ∨ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) (p ∧ q) → (p ∨ q) (¬p ∧ ¬q ∧ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) (¬p ∧ q ∧ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q)

(p ∧ q) → (p ∨ q) (¬p ∨ ¬q ∨ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) (¬p ∧ q ∧ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) Reason: For both the second and third conjuncts to be true, either p or q is true and the other false. This implies that the first conjunct is true. Hence, it is satisfiable. (p ∧ q) → (p ∨ q) Reason: This proposition is a tautology because if (p ∧ q) is true, then (p ∨ q) is also true. So it is obviously satisfiable. (¬p ∧ ¬q ∧ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) Reason: For both the second and third conjuncts to be true, either p or q is true and the other false. But then the first conjunct is false. Hence, no assignment of truth values makes the compound proposition true. (¬p ∧ q ∧ ¬r) ∧ (p ∨ q) ∧ (¬p ∨ ¬q) Reason: For both the second and third conjuncts to be true, either p or q is true and the other false. So, if p is false, q is true, and r is false, the compound proposition is true.

NEGATION of: All professors have eaten lunch at the student center. All professors have eaten lunch at the student center. drop zone empty. There is a student who has not been to Mexico. There is a student who has not been to Mexico. drop zone empty. There is a student who has received a perfect test score. There is a student who has received a perfect test score. drop zone empty. Every student likes pizza.

All professors have eaten lunch at the student center. matches Choice, There is a professor who has not eaten lunch at the student center. There is a professor who has not eaten lunch at the student center. There is a student who has not been to Mexico. matches Choice, Every student has been to Mexico. Every student has been to Mexico. There is a student who has received a perfect test score. matches Choice, Every student received a test score that is not perfect. Every student received a test score that is not perfect. Every student likes pizza. matches Choice, There is a student who does not like pizza. There is a student who does not like pizza.

Consider the following vocabulary: L(x) denoting "x is in the library," D(x) "x is written in Danish," T(x) "x is written in Tamil," and N(x) is "x is a novel." Assume the domain of all variables is all books. Match the English sentence on the left with the correct logical expression on the right. Instructions All the books in the library are written in Danish or Tamil. All the books in the library are written in Danish or Tamil. drop zone empty. There are some books in the library written in Danish or Tamil. There are some books in the library written in Danish or Tamil. drop zone empty. Every book written in Danish or Tamil is in the library. Every book written in Danish or Tamil is in the library. drop zone empty. All books in the library written in Danish are novels. All books in the library written in Danish are novels. drop zone empty. Every book in the library that is not a novel is written in Tamil. Every book in the library that is not a novel is written in Tamil. drop zone empty. The library has all books not written in Tamil or Danish.

All the books in the library are written in Danish or Tamil. matches Choice, ∀x (L(x) →(D(x) ∨T(x))) ∀x (L(x) →(D(x) ∨ T(x))) There are some books in the library written in Danish or Tamil. matches Choice, ∃x((L(x) ∧ (D(x) ∨T(x))) ∃x ((L(x) ∧ (D(x) ∨ T(x))) Every book written in Danish or Tamil is in the library. matches Choice, ∀x ((D(x) ∨T(x)) →L(x)) ∀x ((D(x) ∨ T(x)) → L(x)) All books in the library written in Danish are novels. matches Choice, ∀x ((L(x) ∧D(x)) →N(x)) ∀x ((L(x) ∧ D(x)) → N(x)) Every book in the library that is not a novel is written in Tamil. matches Choice, ∀x ((L(x) ∧ ¬N(x)) →T(x)) ∀x ((L(x) ∧ ¬N(x)) → T(x)) The library has all books not written in Tamil or Danish.

Which of the following English sentences represents ∀x P(x), where P(x) is the statement that "x has more than 1 GB RAM" and the domain is all of the computers at a university? (Select all that apply.) Multiple select question. Some computers have more than 1 GB RAM. Each computer has more than 1 GB RAM. Most of the computers have more than 1 GB RAM All the computers have more than 1 GB RAM.

All the computers have more than 1 GB RAM. Each computer has more than 1 GB RAM. Some computers have more than 1 GB RAM. Reason: There still may be computers without more than 1 GB RAM, and therefore, ∀x P(x) might not hold. Each computer has more than 1 GB RAM. Reason: One way of expressing the universal quantifier in English is to use the word each. Most of the computers have more than 1 GB RAM Reason: Even if most of the computers have more than 1 GB RAM, there still can be some computers that have less. Therefore, it would not be the case that ∀x P(x). All the computers have more than 1 GB RAM. Reason: The use of the word all is a standard way of expressing the universal quantifier in English.

Which of these existential quantifications are true, where the domain of x is the positive integers? Multiple select question. ∃x (x > 0) ∃x (x2 > x) ∃x (x2 < x) ∃x (x < 0)

a,b

Which of the following are valid logical equivalences? Multiple select question. ¬(p ∧ q) ≡ ¬q ∨ ¬p p ∧ F ≡ p ¬(p ∧ q) ≡ ¬q ∧ ¬p p ∨ (p ∧ q)≡ p p ∨ (q ∧ r) ≡ (p ∨ q) ∧ r

a,c ¬(p ∧ q) ≡ ¬q ∨ ¬p Reason: This is one of De Morgan's laws p ∧ F ≡ p Reason: The left-hand side is always false, while the right-hand side can be either true or false, depending on the truth value of p. The equivalence p ∧ T ≡ p is valid. ¬(p ∧ q) ≡ ¬q ∧ ¬p Reason: Note that if p is false and q is true, the left-hand side is true, but the right-hand side is false. The equivalence ¬(p ∧ q) ≡ ¬q ∨ ¬p is valid. p ∨ (p ∧ q)≡ p Reason: This is one of the absorption laws. p ∨ (q ∧ r) ≡ (p ∨ q) ∧ r Reason: The left-hand side is true if p is true and both q and r are false. But the right-hand side is then false. The equivalence p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) is valid. This is one of the distributive laws.

Which of the following are correct ways of determining if two compound propositions p and q are equivalent. (Select all that are correct.) Construct a truth table for p → q. If it is a tautology, then p and q are equivalent. Construct a truth table for p ↔ q. If it is a tautology, then p and q are equivalent. Construct a truth table for both of the compound propositions p and q. If the two are true in the same rows, then p ≡ q. Construct a truth table for (p → q) ∧ (q → p). If the proposition is tautology, then p and q are equivalent.

b,c,d Construct a truth table for p → q. If it is a tautology, then p and q are equivalent. Reason: It is not enough for p → q to be a tautology. Rather we must determine if p ↔ q is a tautology. Construct a truth table for p ↔ q. If it is a tautology, then p and q are equivalent. Reason: p ↔ q is a tautology if and only if p ≡ q. Construct a truth table for both of the compound propositions p and q. If the two are true in the same rows, then p ≡ q. Reason: This is the definition of equivalence. Construct a truth table for (p → q) ∧ (q → p). If the proposition is tautology, then p and q are equivalent. Reason: (p → q) ∧ (q → p) and p ↔ q are equivalent and are tautological exactly when p and q are equivalent.

tautologies Y/N p → p p ¬p ∨ p ¬p → p

p → p ¬p ∨ p p → p Reason: The only way for an implication to have a truth value of F is for the antecedent to have the truth value T and the consequence to have the truth value F. Since the antecedent and the consequence are both the same, this cannot happen here. Therefore, the sentence is always true. p Reason: This sentence can be either true or false, depending on the truth value of p. ¬p ∨ p Reason: If p is true, by the truth table for disjunction, the compound expression ¬p ∨ p is true. On the other hand, if p is false, ¬p ∨ p is true. Therefore, the sentence is always true. ¬p → p Reason: When p is false, this is false.

Let P(x, y) denote "x > y," where the domain for both variables is the set of integers. For which values of x, y is P(x,y) true? Multiple select question. x = 3, y = 0 x = 250, y = 5 x = 250, y = 250 x = 5, y = 7

x = 3, y = 0 x = 250, y = 5