SE 3306 Exam 1

p = "The mushroom is poisonous." r = "The mushroom is red." w = "The mushroom has white spots." y = "The mushroom is yellow." Provide a propositional statement for the following English sentence: If the mushroom is red and has white spots, or if it is yellow, then it is poisonous.

((r ∧ w) ∨ y) → p

u = "The animal is a unicorn." m = "The animal is a mammal." h = "The animal has horns." i = "The animal is immortal." Provide the compound proposition to the following English sentence: The animal is a unicorn if it has horns, is immortal, and is a mammal.

(h ∧ i ∧ m) → u

p = "The mushroom is poisonous." r = "The mushroom is red." w = "The mushroom has white spots." y = "The mushroom is yellow." Provide a propositional statement for the following English sentence: The mushroom is red and has white spots, but it is not poisonous.

(r ∧ w) ∧ ¬p

p = "The mushroom is poisonous." r = "The mushroom is red." w = "The mushroom has white spots." y = "The mushroom is yellow." Provide a propositional statement for the following English sentence: The mushroom is either red and has white spots or is is yellow.

(r ∧ w) ∨ y

u = "The animal is a unicorn." m = "The animal is a mammal." h = "The animal has horns." i = "The animal is immortal." Provide the compound proposition to the following English sentence: If the animal is a unicorn and has horns, then it is a mammal and is immortal.

(u ∧ h) → (m ∧ i)

u = "The animal is a unicorn." m = "The animal is a mammal." h = "The animal has horns." i = "The animal is immortal." Provide the compound proposition to the following English sentence: The animal is either a unicorn and has horns, or it is immortal and a mammal.

(u ∧ h) ∨ (i ∧ m)

Taking into account rules of precedence, provide an expression that is equivalent to ¬p ∧ r → q ∧ s

(¬p ∧ r) → (q ∧ s)

p = "Alex worked for at least 10 hours on the assignment." q = "Sruthi worked for less than 5 hours." What is the negation of the sentence "lex worked for at least 10 hours on the assignment, but Sruthi worked for less than 5 hours."?

1. Alex worked for less than 10 hours on the assignment, or Sruthi worked for at least 5 hours on the assignment. 2. Alex did not work for at least 10 hours on the assignment, or Sruthi worked for at least 5 hours on the assignment.

Suppose p is "Bill has an Android phone" and q is "Sruti does not have an Android phone". Provide English sentences that represent p ∧ q

1. Bill has an Android phone but Sruti does not have an Android Phone. 2. Bill has an Android phone, and Sruti does not have an Android phone.

Provide the correct English statement for the following proposition: ∃x P(x)

1. For some x, P(x) 2. There exists an x such that P(x) 3. There is at least one x such that P(x) 4. There is an x such that P(x)

p = "Ghada's cell phone has less than 16 GB memory." q = "Ghada's laptop has more than 8 GB of memory." "Ghada's cell phone has less than 16 GB of memory or her laptop has more than 8 GB of memory." is represented by p V q. Provide the correct English Translation using De Morgan's Laws of -(p V q).

1. Ghada's cell phone has 16 GB or more of memory, and her laptop has 8 GB of memory or less. 2. Ghada's cell phone does not have less than 16 GB of memory, and her laptop does not have more than 8 GB of memory.

Provide the correct English statement for the following proposition: ∀x P(x)

1. P(x) is true for each x 2. For an arbitrary x P(x) 3. For every x P(x) 4. For all x P(x)

Provide two English sentences that represents p ∧ q where p is "The cat is on the mat" and q is "The dog is outside."

1. The cat is on the mat, but the dog is outside. 2. The cat is on the mat, and the dog is outside.

In which of the rows of the truth table is the compound proposition (p ∧ r) → ¬(q ∨ p) true?

1. The row where both p and q are true, but r is false. 2. The row where p, q, and r are all false.

p = "The toy is lightweight." q = "The toy floats." Provide examples of representation of the biconditional p ⟷ q according to the above statements.

1. The toy is lightweight iff the toy floats. 2. If the toy is lightweight then the toy floats, and conversely. 5. The toy being lightweight is necessary and sufficient for the toy floating.

Consider the English Sentence "If you do not read the book, you will be unable to pass the exam." The original implication of the sentence is:

A necessary condition for not reading the book is being unable to pass the exam.

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? A. x = 250, y = 5 B. x = 3. y = 0 C. x = 250, y = 250 D. x = 5. y = 7

A, B

For which predicates P is the statement ∀x P(x) true, where the domain is the positive integers? A. P(x) is the statement "x > 0" B. P(x) is the statement "x > 2" C. P(x) is the statement "x^2 >= x" D. P(x) is the statement "x^2 > x"

A, C

Example of a Proposition

Aristotle was born in Northern Greece

Which law does this represent? (p ∧ q) ∧ r = p ∧ (q ∧ r)

Associative Law

For which of these universes is ∃x (x^3 <= x^2) true? A. {2,3,4,5} B. {0,2,3,4,5} C. All negative integers D. All positive integers greater than 1 E. All positive integers

B, C, E

Which is the contradiction of ¬(p → p)? A. p V p B. -p C. -p ∧ p

C

Which law does this represent? p ∧ q = q ∧ p

Commutative Law

Which law does this represent? -(p ∧ q) = -q ∧ -p

De Morgan's Law

Which law does this represent? p ∧ (q ∧ r) = (p ∧ q) V (p ∧ r)

Distributive Law

s: "The transaction has a shared lock on the database element." e: "The transaction has an exclusive lock on the element." m: "The transaction can modify the element." r: "The transaction can read the element." Write the English statement for the following propositional logic statement: e → (r ∧ m)

If the transaction has an exclusive lock on the element, then it can modify the element and read the element.

Consider the English Sentence "If you do not read the book, you will be unable to pass the exam." The converse of the sentence is:

If you are unable to pass the exam, you did not read the book.

Consider the English Sentence "If you do not read the book, you will be unable to pass the exam." The Contrapositive of the sentence is:

If you are unable to pass the exam, you have read the book.

Consider the English Sentence "If you do not read the book, you will be unable to pass the exam." The Inverse of the sentence is:

If you read the book, you will be able to pass the exam.

Provide the Negation for the following English Statement: Sundari has a Mac with 8 GB of memory.

It is not the case that Sundari has a Mac with 8 GB of memory.

Suppose p is "Karthik has my copy of the AI textbook" and q is "Sruti has my copy of the AI textbook". Provide an English sentence for: p ∧ q

Karthik and Sruti have my copies of the AI textbook.

Suppose p is "Karthik has my copy of the AI textbook" and q is "Sruti has my copy of the AI textbook". Provide an English sentence for: ¬p ∧ q

Karthik does not have my copy of the AI textbook, but Sruti does have my copy.

Suppose p is "Karthik has my copy of the AI textbook" and q is "Sruti has my copy of the AI textbook". Provide an English sentence for: p V q

Karthik has my copy of the AI textbook, or Sruti has my copy of the AI textbook. Karthik or Sruti has my copy of the AI textbook.

Provide the Negation for the following English Statement: Monty Hall is at least 6 feet tall.

Monty Hall is less than 6 feet tall.

Provide the Negation for the following English Statement: Monty Hall is 6 feet tall.

Monty Hall is not 6 feet tall.

Which law does this represent? p ∧ -p = F

Negation Law

Is this a proposition? Did Bill go to the store yesterday?

No

Is this a proposition? Sit down!

No

Is this a proposition? x + 2 = y

No

Provide the precondition and postcondition that verifies the following code's correctness. t := x x := 2y y := 2t

Precondition: P(x,y) is the statement "x = a and y = b" Postcondition: Q(x,y) is the statement "y = 2a and x = 2b"

Provide the precondition and postcondition that verifies the following code's correctness. t := x x := y + 2 y := 2t

Precondition: P(x,y) is the statement "x = a and y = b" Postcondition: Q(x,y) is the statement "y = 2a and x = b + 2"

Provide the precondition and postcondition that verifies the following code's correctness. t := x x := y + 2 y := t

Precondition: P(x,y) is the statement "x = a and y = b" Postcondition: Q(x,y) is the statement "y = a and x = b + 2"

Provide the precondition and postcondition that verifies the following code's correctness. t := x x := y^2 y := t

Precondition: P(x,y) is the statement "x = a and y = b" Postcondition: Q(x,y) is the statement "y = a and x = b^2"

Provide the Negation for the following English Statement: Sundari has a Mac with less than 8 GB of memory.

Sundari has a Mac with at least 8 GB of memory.

Provide a domain for the variable x that will make the statement true. ∀x (x^2 < -1)

The Empty Set

Provide a domain for the variable x that will make the statement true. ∀x (x < 0)

The Negative Integers

Provide a domain for the variable x that will make the statement true. ∀x (x > 0)

The Positive Integers

Provide a domain for the variable x that will make the statement true. ∀x (x^2 = 1)

The set (-1,1)

s: "The transaction has a shared lock on the database element." e: "The transaction has an exclusive lock on the element." m: "The transaction can modify the element." r: "The transaction can read the element." Write the English statement for the following propositional logic statement: (e ∨ s) ∧ m

The transaction either has an exclusive lock or a shared lock on the element and it can modify the element.

s: "The transaction has a shared lock on the database element." e: "The transaction has an exclusive lock on the element." m: "The transaction can modify the element." r: "The transaction can read the element." Write the English statement for the following propositional logic statement: s ⟷ (r ∧ ¬m)

The transaction has a shared lock on the element if and only if it can read the element but cannot modify the element.

s: "The transaction has a shared lock on the database element." e: "The transaction has an exclusive lock on the element." m: "The transaction can modify the element." r: "The transaction can read the element." Write the English statement for the following propositional logic statement: s → (r ∧ ¬m)

The transaction has a shared lock on the element only if it can read the element but cannot modify it.

How do you determine if two compound propositions p and q are equivalent?

Two ways: 1. construct a truth table for both compound propositions p and q, if the two are true in the same rows, then p=q. 2. Construct a truth table (p->q) and (q-> p). If the proposition is a tautology, then p and q is equivalent.

p = "We will name our baby Mary." q = "We will name our baby Phillip." What is p ⊕ q ?

We will name our baby Mary or Phillip, but not both.

Is this a proposition? 1 + 1 = 2

Yes

Is this a proposition? 1 + 1 = 3

Yes

Is this a proposition? The cat is on the mat.

Yes

p = "Joe went to France." q = "Jessica went to England." "Joe went to France and Jessica went to England" is the _______________ of p and q.

conjunction

Consider the sentence "If you came before 4:00, we will get there in time for dinner." The sentence "If we did not get there in time for dinner, you did not come before 4:00" is the _______ of the original sentence.

contrapositive

Consider the sentence "If you came before 4:00, we will get there in time for dinner." The sentence "If we get there in time for dinner, you came before 4:00" is the _______ of the original sentence.

converse

p = "Joe went to France." q = "Jessica went to England." "Joe went to France or Jessica went to England" is the _______________ of p and q.

disjunction

Consider the sentence "If you came before 4:00, we will get there in time for dinner." The sentence "If you did not come before 4:00, we did not get there in time for dinner" is the _______ of the original sentence.

inverse

The sentence "Zaid does not have a newborn baby" is the _______ of the statement "Zaid has a newborn baby."

negation

Does the following sentence make use of universal quantification? Some integers have an opposite.

no

Does the following sentence make use of universal quantification? There are birds that cannot fly.

no

Does the following sentence use "or" with an exclusive meaning? I'm available on Fridays after 4:00 or on Wednesdays before 2:00.

no

Does the following sentence use "or" with an exclusive meaning? No service without shirt or shoes.

no

Does the following sentence use "or" with an inclusive meaning? When tossing a coin, you will get a head or a tail.

no

Does the following sentence use "or" with an inclusive meaning? You may eat dinner this evening in the cafeteria or in the dining hall.

no

Is the following statement a tautology? p

no

Is the following statement a tautology? ¬p → p

no

Is this compound proposition satisfiable? (-p ∧ -q ∧ -r) ∧ (p ∨ q) ∧ (-p ∨ -q)

no

Provide the truth values of the propositional variables that make the following expression true: ¬(p ∨ q)

p = F and q = F

Provide the truth values of the propositional variables that make the following expression true: p ∨ (q ∧ r)

p = T

p = "The mushroom is poisonous." r = "The mushroom is red." w = "The mushroom has white spots." y = "The mushroom is yellow." Provide a propositional statement for the following English sentence: If the mushroom is poisonous, it is red and has white spots, or it is yellow.

p → ((r ∧ w) ∨ y)

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: Passing the exam implies that the student has read the book.

p → q

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: The student having passed the exam is sufficient for the student having read the book.

p → q

p = "The student has read the book." q = "The student passed the exam." Provide the propositional logic for the following English statement: The student has read the book and has not passed the exam.

p ∧ ¬q

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: The student has passed the exam or the student has read the book.

p ∨ q

Provide the truth values of the propositional variables that make the following expression true: ¬p ∨ q

q = T

p = "The student has read the book." q = "The student passed the exam." Provide the propositional logic for the following English statement: The student has read the book if the student passes the exam.

q → p

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: Having read the book implies that the student will pass the exam.

q → p

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: The student having passed the exam is necessary for the student having read the book.

q → p

Provide the truth values of the propositional variables that make the following expression true: (p → q) → r

r = T

u = "The animal is a unicorn." m = "The animal is a mammal." h = "The animal has horns." i = "The animal is immortal." Provide the compound proposition to the following English sentence: Having horns, being a mammal, and being immortal is necessary and sufficient for the animal being a unicorn.

u ⟷ (m ∧ i ∧ h)

Does the following sentence make use of universal quantification? All of the students are present.

yes

Does the following sentence make use of universal quantification? Every number is interesting.

yes

Does the following sentence make use of universal quantification? For an arbitrary integer, its square is positive.

yes

Does the following sentence use "or" with an exclusive meaning? Carol is either in the attic or the garage.

yes

Does the following sentence use "or" with an exclusive meaning? George Boole was either born in England or Ireland.

yes

Does the following sentence use "or" with an inclusive meaning? To enjoy winter, you have to know how to ski or how to skate.

yes

Does the following sentence use "or" with an inclusive meaning? You cannot go on the ride if you are under 4 feet tall or if you weigh over 300 pounds.

yes

Is the following statement a tautology? p → p

yes

Is the following statement a tautology? ¬p ∨ p

yes

Is this compound proposition satisfiable? (-p ∧ q ∧ -r) ∧ (p q) ∧ (-p -q)

yes

Is this compound proposition satisfiable? (-p ∨ -q ∨ -r) ∧ (p ∨ q) ∧ (-p ∨ -q)

yes

Is this compound proposition satisfiable? (p ∧ q) → (p ∨ q)

yes

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: The student not having passed the exam is sufficient for the student not having read the book.

¬p → ¬q

p = "The student has read the book." q = "The student passed the exam." Provide the propositional logic for the following English statement: Not having read the book is necessary and sufficient for passing the exam.

¬p ⟷ q

p = "The student has read the book." q = "The student passed the exam." Provide the propositional logic for the following English statement: If it is not the case that the student reads the book then it is not the case that the student passes the exam, and conversely.

¬p ⟷ ¬q

p = "The student passed the exam." q = "The student has read the book." Provide the propositional logic for the following English statement: The student has not read the book and has not passed the exam.

¬q ∧ ¬p