Intro to Discrete Maths

Which rule of inference is the basis of "All cats are furry. Felix is a cat. Therefore, Felix is Furry."

Universal instantiation

An example of Associative Law is

(p ^ q) ^r = p ^(q^r)

An example of Domination law is

p ^F = F

∀x∀y((x<0) ^ (y < 0) → (xy > 0)) in English is

For any two negative numbers, the product is positive

"Never Copy Ideas" is used to remember?

"Negation, Conjunction, Implication"

Match the quantified statement having the domain consisting on the integers from 3 to 5 to it's corresponding propositional logic. 1. ∃x(x > 4) 2. ∃x(4 > x) 3. ∀x(x > 4) 4. ∀x(4 > x)

1. (3 > 4) v (4 > 4) v (5 > 4) 2. (4 > 3) v (4 > 4) v (4 > 5) 3. (3 > 4) ^ (4 > 4) ^ (5 > 4) 4. (4 > 3) ^ (4 > 4) ^ (4 > 5)

Which of these would a proof of contraposition be a better approach than a direct approach? 1. If 3n+2 is odd, then n is odd 2. Prove that if 1/x is irrational then x is irrational 3. If a is odd, and b is odd, then ab is odd 4. If n is odd, then n^3+1 is even

1. If 3n+2 is odd, then n is odd 2. Prove that if 1/x is irrational then x is irrational

Let P(x, y) denote the statement "x is taller than y." Suppose Bob is 5'7, Jill is 5'3, Azzam is 5'11, and Keya is 6'. Which of the following is True? 1. P(Azzam, Jill) 2. P(Keya, Azzam) 3. P(Jill, Keya) 4. P(Jill. Bob) 5. P(Azzam, Bob)

1. P(Azzam, Jill) 2. P(Keya, Azzam) 5. P(Azzam, Bob)

What values of the propositional variables make the expression true? 1. ¬p v q 2. ¬ (p v q) 3.(p → q) → r 4. p v (q ^r)

1. q= t 2. p = F and q = F 3. r= T 4. p = T

For which of these universes is ∃x(x^3 =< x^2) true? 1. {2,3,4,5} 2. All Negative Integers 3. {0,2,3,4,5} 4. All positive integers 5. All positive integers greater than 1

2. All Negative Integers 3. {0,2,3,4,5} 4. All positive integers

Premises "If the weather is good, people go to the beach," "If people go to the beach, they are happy," and "The weather is good" lead to the conclusion "People are happy." Let g be the proposition "the weather is good," Let b be the proposition "people go to the beach," Let h be the proposition "People are happy." Which of these is a Modus ponens using two premises? 1. g 2. b 3. h 4. g → b 5. b → h

2. b

For which predicts P is the statement ∀xP(x) true, where the domain is the positive integers? 1. P(x) is the statement "x^2 >x" 2. P(x) is the statement "x >2" 3. P(x) is the statement "x > 0" 4. P(x) is the statement "x^2 >= x"

3. P(x) is the statement "x > 0" 4. P(x) is the statement "x^2 >= x"

Which rule of inference is the basis of "It is snowing. Therefore it is snowing or it is sunny"?

Addition

p→ ¬p is an example of

Contingency. It is false when p is true, and true when p is false.

A Proposition is a statement. "You are walking" is a proposition.

True. A proposition is a statement that can be proven true or false. "Do not pass go" is NOT a proposition

Expressions with nested quantifiers expressing statements in English can be quite complicated. The first step in translating such an expression is to write out what the quantifiers and predicates in the expression mean. The next step is

to express this meaning in a simpler sentence.

(p∨q)∧(¬r) is the conjunction of p∨q and ¬r. However, to reduce the number of parentheses, we specify that the negation operator is applied before all other logical operators. This means that ¬p∧q is the conjunction of

¬p and q, namely, (¬p)∧q, not the negation of the conjunction of p and q, namely ¬(p∧q)

The Negation of p, denoted by ¬p, is the statement.

Let "P" be proposition. The Negation of p, denoted by ¬p, is the statement. "It is not the case that p." The proposition ¬p is read "not p." The truth value of the negation p, ¬p, is the opposite of the truth value of p.

Let p and q be propositions. The biconditional statement p↔q is the proposition "p if and only if q." The biconditional statement p↔q is true when

p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications.

An example of Commutative Law is

p^q = q ^ p

An example of De Morgan's Law is

¬(p v q) = ¬q ^ ¬p

Many mathematical statements involve multiple quantifications of propositional functions involving more than one variable. It is important to note that

the order of the quantifiers is important, unless all the quantifiers are universal quantifiers or all are existential quantifiers.

Let p be "Ghada's Cell phone has less than 16GB memory" and q be "Ghada's laptop has more than 8gb of Memory." Then the Sentence "Ghada's cell phone has less than 16ghb memory or her laptop has more than 8GB memory" is represented as p V q. What is the correct English translation using DeMorgan's law of ¬(p V q)?

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

There is an island in which certain inhabitants called "Knights" always tell the truth, and other called "Knaves" always lie. We assume that all of the inhabitants are either knights or knaves. Who is Who? 1. "A says If I am a knight, then so is B." 2. "B says If A is a knight then I am a Knave." 3. "A says I am a knave, but B isn't." 4. "A says Either I am a knave or else two plus two equals five. B says nothing."

1. "A is a Knight and B is a Knight." 2. "A is a knave and B is a Knight." 3. "A is a knave and B is a Knave." 4. " There is no valid assignment of types as the statements are contradictory.

Consider the English Sentence "If you do not read the book, you will be unable to pass the exam." 1. If you are unable to pass the exam, you did not read the book. 2. If you are able to pass the exam, you have read the book. 3. If you read the book, you will be able to pass the exam. 4. A necessary condition for not reading the book is being unable unable to pass the exam.

1. Converse 2. Contrapositive 3. Inverse 4. Original Implication

What belongs under the proposition ∃xP(x) ? 1. There Exists an x such that P(x) 2. For some x, P(x) 3. There is an x such that P(x) 4. There is at least one x such that P(x) 5. For an arbitrary x P(x) 6. For all x P(x) 7. P(x) is true for each x 8. For every x p(x)

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

If p is "Joe went to France" and q is "Jessica went to England," then "Joe went to France or Jessica went to England" is the ____ of p and q. On the other hand "Joe went to France and Jessica went to England" is the ____ of p and q.

1. disjunction 2. conjunction

Premises "If the weather is good, people go to the beach," "If people go to the beach, they are happy," and "The weather is good" lead to the conclusion "People are happy." Let g be the proposition "the weather is good," Let b be the proposition "people go to the beach," Let h be the proposition "People are happy." Which of these is the Premise? 1. g 2. b 3. h 4. g → b 5. b → h

1. g 4. g → b 5. b → h

Which of these can easily be proven by a direct proof? 1. If 5n+2 is even where n is an integer, then n is even 2. If n is odd, then n^3+1 is even 3. If n =(ab)^2 where a and b are positive integers, then n >=a^2 and n >= b^2 4. If n is a perfect square where n is a positive integer, then n+1 is not a perfect square.

2. If n is odd, then n^3+1 is even 3. If n =(ab)^2 where a and b are positive integers, then n >=a^2 and n >= b^2 4. If n is a perfect square where n is a positive integer, then n+1 is not a perfect square.

In which of these expressions is the variable x free? 1. ∀xP(x) v ∃x(q(x)→P(x)) 2. ∃xP(y) ^ Q(x) 3. ∀x(P(x) ^ Q(y)) 4. ∃x(p(x) ^ q(y))

2. ∃xP(y) ^ Q(x) 3. ∀x(P(x) ^ Q(y))

Translate "All web pages owned by Ernest have a link to a page that is about chess" into predicate logic. Assume that the universe of discourse of all variable is all Web Pages. For the predicates let L(x,y) be "x links to y," O(x,y) be "x owns y" and A(x,y) be "x is about y." 1. ∀x∃y(O(Earnest,x) ^ (L(x,y) ^ A(y, chess))) 2. ∃x∃y(O(Earnest,x)^ (L(x,y) ^ A(y, chess))) 3. ∀x∃(O(Earnest,x)→ (L(x,y) ^ A(y, chess))) 4. ∀x∀y(O(Earnest,x)→ (L(x,y) ^ A(y, chess)))

3. ∀x∃(O(Earnest,x)→ (L(x,y) ^ A(y, chess)))

These are the steps of a proof by contraposition of "If n is an integer and 5n+3 is odd, then n is even." Match the step to the explanation 1. Assume that n is not even, that is, that it is odd 2. Then n=2k+1 for some integer k 3. 5n+3=5(2k+1)+3=10k+8=2(5k+4) 4. Hence, 5n+3 is even

1. Negation of the conclusion 2. Definition of an odd integer 3. Arithmetic 4. Definition of an even number

p<-> ¬p is an example of

Contradiction. This is always false because p and ¬p have opposite truth values.

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 will have gotten there before 4:00" is the...

Converse of the sentence

Is the Notation for the negation operator standardized?

No, There are others.

Besides the phrase "there exists," we can also express existential quantification in many other ways, such as by using the words "for some," "for at least one," or "there is." The existential quantification ∃xP(x) is read as

"There is an x such that P(x)," "There is at least one x such that P(x)," or "For some xP(x)."

Assume that P(x) is the statement "x =< x^3." For which of these domains is ∀xP(x) true? 1. All negative integers 2. -2, -1, 0, 1, and 2 3. -1, 0, 1 4. All Positive Integers 5. All integers

3. -1, 0, 1 4. All Positive Integers

Match the English sentence with the correct Statement in propositional logic, where the proposition p is "The Student passed the exam," and q is "The Student has read the book." It is either "p→q," "q→p," "p^¬q," or "None of these."

"The Student passed the exam, but has not read the book" is p^¬q "The student has passed the exam unless the student has not read the book." is q→p "The Student has passed the exam only if the student has read the book." is p→q "The student either does not read the book or does not pass the exam." None of the choices.

1. p→q 2. p ↔q 3. ¬(p↔q) 4. (p→r) ^ (q→r) 5. (p→q) V (p→r) 6. (p→r) V (q→r)

1. ¬q→¬p 2. (p→q) ^ (q → p) 3. p ↔ ¬q 4. (p v q) → r 5. p → (q V r) 6. (p ^ q) → r

An example of Double Negation Law is

¬(¬p)=p

Which of the following are true? The domain of all variables is the integers. 1. ∀x∃y(xy=1) 2. ∀x∀y(xy=yx) 3. ∀x∃y(x=2y) 4. ∃x∀y(xy=y)

2. ∀x∀y(xy=yx) 4. ∃x∀y(xy=y)

Which of the following English sentences represents ∃xQ(x) where Q(x) is the statement that X is a computer science student and the domain is all the students in the class? 1. Each students in the class is a computer science student 2. Most of the students in this class are computer science students 3. At least one student in this class is a computer science student. 4. Each student in the class is a computer science major.

3. At least one student in this class is a computer science student. 4. Each student in the class is a computer science major.

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

inverse

it is an accepted rule that the conditional and biconditional operators, → and ↔, have lower precedence than the conjunction and disjunction operators, ∧ and ∨. Consequently, p→q∨r means

p→(q∨r) rather than (p→q)∨r and p∨q→r means (p∨q)→r rather than p∨(q→r)

We define the bitwise OR, bitwise AND, and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR, AND, and XOR of the corresponding bits in the two strings, respectively. We use the symbols ∨, ∧, and ⊕ to represent

the bitwise OR, bitwise AND, and bitwise XOR operations, respectively.

Generally, an implicit assumption is made that all domains of discourse for quantifiers are nonempty. Note that if the domain is empty, then ∀xP(x) is true for any propositional function P(x) because

there are no elements x in the domain for which P(x) is false

the conjunction operator takes precedence over the disjunction operator, so that p∨q∧r means p∨(q∧r) rather than (p∨q)∧r and p∧q∨r means (p∧q)∨r rather than p∧(q∨r).

Because this rule may be difficult to remember, we will continue to use parentheses so that the order of the disjunction and conjunction operators is clear.

Express the negation of the statement ∀x∃y(xy=1)∀x∃y(xy=1) so that no negation precedes a quantifier.

By successively applying De Morgan's laws for quantifiers, we can move the negation in ¬∀x∃y(xy=1) inside all the quantifiers. We find that ¬∀x∃y(xy=1) is equivalent to ∃x¬∃y(xy=1), which is equivalent to ∃x∀y¬(xy=1). Because ¬(xy=1) can be expressed more simply as xy≠1, we conclude that our negated statement can be expressed as ∃x∀y(xy≠1)

What are the equivalents? 1. ¬∃xP(x) 2. ¬∀xP(x) 3. ¬∀x→P(x) 4. ¬∃x →P(x)

1. ∀x¬P(x) 2. ∃x¬P(x) 3. ∃xP(x) 4. ∀xP(x)

What is it's logical equivalent to it's negation, where no negation operator is to the left of a quantifier in it's scope. 1. ∀x∃y(P(x,y)^Q(y,z)) 2. ∃x∀y(P(x,y)^Q(y,z)) 3. ∃x∀y(P(x,y) -> Q(y,z)) 4.∀x∃y(¬P(x,y) v ¬Q(y,z))

1. ∃x∀y(¬P(x,y) V ¬Q(y,z)) 2. ∀x∃y(¬P(x,y) V ¬Q(y,z)) 3. ∀x∃y(P(x,y) ^ ¬Q(y,z)) 4. ∃x∀y(P(x,y) ^ Q(y,z))

Fred is a graduate student and is from Idaho. Graduate students are students. All students are hard working. Everyone from Idaho likes potatoes. Conclude that Fred is hard working and likes potatoes. Let G(x) denote "x is a graduate student," I(x) denote "x is from Idaho," S(x) denote "x is a student," H(x) denote "x is hard working," and P(x) denote "x likes potatoes." Then the premises are 1. G(Fred,) 2. I(Fred,) 3. ∀xG(x) →S(x), 4. ∀xS(x)→H(x), 5. ∀xI(x)→P(x). The conclusion is H(Fred) ^ P(Fred.) Match the statements. 6. G(Fred) → S(Fred) 7. S(Fred) 8. S(Fred) → H(Fred) 9. H(Fred) 10. I(Fred) → P(Fred) 11. P(Fred) 12. H(Fred) ^ P(Fred)

6. Universal instantiation from 3 7. Modus ponens from 1 and 6 8. Universal instantiation from 4 9. Modus ponens from 7 and 8 10. Universal instantiation from 5 11. Modus ponens from 2 and 10 12. Conjunction from 9 and 11

Let p and q be propositions. The Conjunction of p and q, denoted by p^q, is the proposition "p and q." The conjunction of p^q is true when

Both p and q are true, and is false otherwise

The rules for negations for quantifiers are called

De Morgan's laws for quantifiers.

Let p and q be propositions. The exclusive or of p and q, denoted by p⊕q (or p XOR q), is the proposition that is true when exactly one of p and q is true and is false otherwise.

Exactly one of p and q is true and is false otherwise.

These sentences express the Biconditional p↔q where p is "The Toy is lightweight" and q is "The toy floats."

"If the Toy is lightweight, then the toy floats, and conversely." "The toy is lightweight iff the toy floats." "The toy being lightweight is necessary and sufficient for the toy floating."

¬p→ ¬p is an example of

Tautology. This is always true because both sides of the implication have the same truth value.

Where one quantifier is within the scope of another

Nested quantifiers

Because conditional statements play such an essential role in mathematical reasoning, a variety of terminology is used to express p→q. You will encounter most if not all of the following ways to express this conditional statement:

"if p, then q" "if p, q" "p is sufficient for q" "q if p" "q when p" "a necessary condition for p is q" "q unless ¬p¬p" "p implies q" "p only if q" "a sufficient condition for q is p" "q whenever p" "q is necessary for p" "q follows from p" "q provided that p"

Note that the statement p↔q is true when both the conditional statements p→q and q→p are true and is false otherwise. That is why we use the words "if and only if" to express this logical connective and why it is symbolically written by combining the symbols → and ←. There are some other common ways to express p↔q: "p is necessary and sufficient for q" "if p then q, and conversely" "p iff q." "p exactly when q."

"p is necessary and sufficient for q" "if p then q, and conversely" "p iff q." "p exactly when q."

Which of these compound propositions are satisfiable? 1. (p ^ q) → (p V q) 2. (¬p ^ ¬q ^ ¬r) ^ (p V q) ^ (¬p v ¬q) 3. (¬p ^ ¬q ^ r) ^ (p v q) ^ (¬p v ¬q) 4. (¬p V ¬q V r) ^ (p v q) ^ (¬p V ¬q)

1. (p ^ q) → (p V q) 2. (¬p ^ ¬q ^ ¬r) ^ (p V q) ^ (¬p v ¬q) 4. (¬p V ¬q V r) ^ (p v q) ^ (¬p V ¬q)

Taking into the rules of precedence, which of the following parenthesized expressions is equivalent to ¬p ^ r → q ^ s 1. (¬p ^ r) → (q ^ s) 2. ¬p ^ (r → (q ^ s)) 3. ¬((p ^ r) → (q ^ s)) 4. ¬p ^( r → q ) ^ s

1. (¬p ^ r) → (q ^ s) The operator ¬ has the highest precedence, next ^ and after that →

What do the bit strings mean? 1. 0100 2. 1110 3. 1010 4. 1011

1. And 2. Or 3. XOR 4. Not And, Or, or XOR of the two strings.

Match the statement to be proved with the correct use of proof by cases. 1. 3^n < 2n^3 for all positive integers n =< 3 2. n^3 =< n^4 for all intergers 3. A cube and a fourth power of distinct positive intergers less than 10 differ more than 2 4. The final digit of a fourth power of an integer is 0,1,5, and 6

1. Verify the statement for n=1, n=2, n=3 2. Split the proof into 3 cases, n=0, n positive, and n negative 3. Check the truth of the statement for all pairs of integers in the appropriate range 4. Reduce this to 10 cases, each covering the appropriate power of a particular integer

What belongs under the proposition ∀xP(x) ? 1. There Exists an x such that P(x) 2. For some x, P(x) 3. There is an x such that P(x) 4. There is at least one x such that P(x) 5. For an arbitrary x P(x) 6. For all x P(x) 7. P(x) is true for each x 8. For every x p(x)

5. For an arbitrary x P(x) 6. For all x P(x) 7. P(x) is true for each x 8. For every x p(x)

How do you go about proving that 9n - 2 is even if and only if n is even? 1. Use a proof by contraposition for each direction 2. One direct proof is sufficient 3. Use direct proofs for both directions 4. It can be done with one proof by contradiction 5. Use a direct proof for one direction and a contraposition for the other

5. Use a direct proof for one direction and a contraposition for the other

Which rule of inference is the basis of "If it rains, the the grass grows. It is raining. Therefore, the grass grows."

Modus ponens

What do these sentences have in common? "No service without shirt or shoes." and "I'm available on Friday's after 4:00 or Wednesday's before 2:00"

They use Or with an inclusive Meaning. They will not serves someone without a shirt, without shoes, or without both shoes and a shirt. She is available on both days, so it is OR.

The following are valid equivalences, true or false? p ↔ q =¬ p ↔ ¬q (p → q) ^ ( p → r) = p →(q V r) p V q = ¬p → q

True

Are the following steps in order of a complete series of logical equivalence? 1. = ¬p v ¬ (¬p v q) 2. = ¬p v (¬¬p ^¬q) 3. = ¬p v (P ^ ¬q) 4. = (¬p v p) ^ ¬p V ¬q) 5. = T ^(¬p v ¬q) 6. (¬p v ¬q) ^ T

Yes

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

contrapositive

The connective or is also used to express an exclusive or. Unlike the disjunction of two propositions p and q, the exclusive or of these two propositions is true when

exactly one of p and q is true; it is false when both p and q are true (and when both are false)

Match the English sentence with the correct Statement in propositional logic, where the proposition p is "The Student passed the exam," and q is "The Student has read the book." It is either "p→q," "q→p," "p^¬q," or "None of these." 1. "Having read the book follows from having passed the exam." 2. "A sufficient condition for the student passing is having read the book." 3. "If the student did not read the book, then the student did not pass the exam." 4. "The student has read the book but did not pass the exam."

1. p → q (The expression "q follows from p" is a way of stating this.) 2. q → p (The expression "A sufficient condition for p is q" is a way of stating this) 3. None of these ( The sentence is of the form ¬q → ¬p) 4. q ^ ¬p (This is the conjunction of ¬p and q. The word But is a way of expressing disjunction.)

Match with it's counter example. 1. Every Integer is greater than it's square 2. If x^2 us rational, then x is rational. 3. If x is a real number, then x^2+1 >= 2x 4. No two perfect squares can differ by 1. 5. No two negative integers can have 1 as their product.

1. -1 is a counter example 2. x = Square root of 2 is a counterexample 3. A theorem 4. 0 and 1 are counter examples 5. -1 and -1 is a counter example

Fill in the blanks. Moving a negation inward over a disjunction changes the disjunction into a ___ and each of the two statements is ____. Moving a negation inward over an existential quantifier changes the quantifier to a ______ quantifier and the rest of the expression is negated. Moving a negation inward over an universal quantifier changes the quantifier to an ____ quantifier and the rest of the expression is negated.

1. Conjunction 2. Negated 3. Universal 4. existential

Consider the English Sentence "If the Exam is open book, then it will be hard." Match the description to the sentence. 1. "If the Exam is Hard, then it will be open book." 2. "If the Exam is not open book, then it will not be hard." 3. "If the exam is not hard, then it will be closed book." 4. "The exam being open book is sufficient for it being hard."

1. Converse 3. Contrapositive 2. Inverse 4. Original Implication

Which of the following English sentences represents ∀xQ(x) where Q(x) is the statement that X is a computer science major and the domain is all the students in the class? 1. Each student in the class is a computer science Major 2. We have some computer science majors in the class 3. Many of the students in the class are computer science majors. 4. Every student in the class is a computer science major.

1. Each student in the class is a computer science Major 4. Every student in the class is a computer science major.

Match the English sentence with the correct Statement in propositional logic, where the proposition p is "The Student passed the exam," and q is "The Student has read the book." It is either "p→q," "q→p," "p V q," or " ¬q ^ ¬p" 1. "Passing the exam implies that the student has read the book." 2. "The student has passed the exam or the student has read the book." 3. "Having read the book implies that the student will pass the exam." 4. "The student has not read the book and has not passed the exam."

1. p→q 2. p V q 3. q→p 4. ¬q ^ ¬p

Match the english sentences with the corresponding compound propositions. The propositions are p "The mushroom is poisonous," r: "The mushroom is red," w: "The mushroom has white spots," and y: "The mushroom is yellow." 1. "If the Mushroom is red and has white spots, or if it is yellow, then it is poisonous." 2. "The Mushroom is red and has white spots, but it is not poisonous." 3. "If the Mushroom is poisonous, it is red and has white spots, or it is yellow." 4. "The Mushroom is either red and has white spots, or it is yellow."

2. (r ^ w) ^ ¬p 3. p → ((r ^ w) V y) 4. (r ^ w) V y 1. ((r ^ w) V y) → p

Which can be proved with a constructive existence proof, without using a computing device? 1. There is a product of two of the three numbers 2^2014-3^1188+5^666, 7^999-11^552+13^333, and 15^366-17^311+19^222 that is positive. 2. There are positive integers whose squares and cubes end with different decimal digits. 3. There is a positive integer that equals the product of all positive integers not exceeding it. 4. For every digit between 0 and 9 inclusive, there is a perfect cube that ends with that digit.

2. There are positive integers whose squares and cubes end with different decimal digits. 3. There is a positive integer that equals the product of all positive integers not exceeding it. 4. For every digit between 0 and 9 inclusive, there is a perfect cube that ends with that digit.

Which is a correct representation of "If every user in the department is logged in, then there must be at least two servers running," in predicate logic. Let the domain of variable x be all users, and the domain of variables y and z be all pieces of computing machinery at the university. Use D(x) for "x is a user in the department," L(x) for "x is logged in," S(y) for "y is a server," and R(y) for "y is running" 1. (∀x(d(x)→L(x))) → (∀y∀z((S(z)) →(y =/ z) ^ R(y) ^R(z))) 2. (∃x(d(x) ^ L(x))) → (∃y∃z((S(z)) ^ (y =/ z) ^ R(y) ^R(z))) 3. (∀x(d(x)→L(x))) → (∃y∃z((S(z)) ^ (y =/ z) ^ R(y) ^R(z))) 4. (∀x(d(x)→L(x))) → (∀y∀z((S(z)) ^ (y =/ z) ^ R(y) ^R(z)))

3. (∀x(d(x)→L(x))) → (∃y∃z((S(z)) ^ (y =/ z) ^ R(y) ^R(z)))

Express these system specifications using the propositions a, c, and s, where a denotes "The browser is set to accept cookies," c denotes "the cookie is stored by the browser," and s denotes "the server sends the cookie." If the server sends the cookie and the browser is set to accept cookies, then the browser will store the cookies. But if the browser is not set to accept cookies, then the browser will not store the cookie.

((a ^ s) → c) ^ (¬a → ¬c)

The last way of expressing the biconditional statement p↔q uses the abbreviation "iff" for "if and only if." Note that p↔q has exactly the same truth value as (p→q)∧(q→p).

(p→q)∧(q→p).

What causes these to be false? 1. ∀x∀yP(x,y) 2. ∀x∃yP(x,y) 3. ∃x∀yP(x,y) 4. ∃x∃yP(x,y)

1. There is a pair x, y for which P(x,y) is false 2. There is an x such that P(x,y) is false for every y 3. For every x there is a y for which P(x,y) is false 4. P(x,y) is false for every pair (x,y)

What nested quantification's match? 1. ∀x∀yP(x,y) 2. ∀x∃yP(x,y) 3. ∃x∀yP(x,y) 4. ∃x∃yP(x,y)

4. There is a pair x, y for which P(x,y) is true 3. There is an x such that P(x,y) is true for every y 2. For every x there is a y for which P(x,y) is true 1. P(x,y) is true for every pair (x,y)

Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the domain of discourse (or the universe of discourse), often just referred to as the domain. Such a statement is expressed using universal quantification. The universal quantification of P(x) for a particular domain is the proposition that asserts that P(x) is true for all values of x in this domain. Note that the domain specifies the possible values of the variable x. The meaning of the universal quantification of P(x) changes when we change the domain. The domain must

always be specified when a universal quantifier is used; without it, the universal quantification of a statement is not defined

Which of the following propositional logics are tautologies? 1. ¬p v p 2. p 3. ¬p → p 4 p → p

1. ¬p v p 4 p → p

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 variable is all books. Find the logical expression for each sentence. 1. All the books in the library are written in Danish or Tamil 2. There are some books in the library written in Danish or Tamil 3. Every book written in Danish or Tamil is in the library 4. All the books in the library written in Danish are novels 5. Every book in the library that is not a novel is written in Tamil 6. The library has all books not written in Danish or Tamil

1. ∀x(L(x)→(D(x) V T(x))) 2. ∃x((L(x) ^ (D(x) V T(x))) 3. ∀x((D(x) V T(x)) → L(x)) 4. ∀x((L(x) ^ D(x)) → N(x)) 5. ∀x((L(x) ^¬ N(x)) → T(x)) 6. ∀x(. ¬ (D(x) v T(x)) → L(x))

What is the predicate calculus translation for the following? 1. There is exactly one student in this class who has a perfect score. 2. All students in the class have a perfect score. 3. At least one student in the class has a perfect score. 4. There is no one in the class with a perfect score. 5. There is exactly one student in this class, and that student has a perfect score.

1. ∃x((C(x) ^ P(x)) ^ ∀y((y =/ x) → ¬(C(x)^P(x)))) 2. ∀x(C(x) → P(x)) 3. ∃x(C(x)^P(x)) 4. ∀x(¬C(x) V ¬P(x)) 5. ∃x((C(x)^P(x)) ^ ∀y(C(y) → (y=x)))

Translating from English to logical expressions becomes even more complex when quantifiers are needed. Furthermore, there can be many ways to translate a particular sentence.

As a consequence, there is no "cookbook" approach that can be followed step by step.

On a game show, you must pick one of two doors. You know that behind each door there is either a new car or a goat. There may be a car behind both doors, a goat behind both doors, or a car behind one door and a goat behind the other. The sign on door 1 says "IN THIS ROOM, THERE IS A CAR, AND IN THE OTHER ROOM THERE IS A GOAT." The sign on door 2 says "IN ONE ROOM THERE IS A CAR, AND IN THE OTHER THERE IS A GOAT." One of the signs is true and the other is false. Which door should you pick, assuming that you would rather have the car than the goat?

Door 2. To see that this choice is correct, let g be the statement that the goat is behind door 1, g2 be the statement that there is a goat behind door 2, c be the statement that there is a car behind door 1, and c2 be the statement that there is a car behind door 2. The sign on door 1 asserts c ^ g2 is true, while the sign on door 2 is (c ^ g2) V (c2 ^g). If the sign on door 1 is true, that is, if c ^ g2, then certainly (c^g2) V (c2 ^ g) is true as well. Since the signs can both not be true, c^g2 must be false and (c^g2) V (c2 ^ g) is true.

The statement "x is greater than 3" has two parts. The first part, the variable x, is the subject of the statement. The second part—the predicate, "is greater than 3"—refers to a property that the subject of the statement can have. We can denote the statement "x is greater than 3" by P(x), where P denotes the predicate "is greater than 3" and x is the variable. The statement P(x) is also said to be

the value of the propositional function P at x. Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value

Many mathematical statements assert that there is an element with a certain property. Such statements are expressed using existential quantification. With existential quantification, we form a proposition that is true if and only if P(x) is

true for at least one value of x in the domain.

Let p and q be propositions. The disjunction of p and q, denoted by p∨q, is the proposition "p or q." The disjunction p∨q is false when

both p and q are false and is true otherwise.

We can form some new conditional statements starting with a conditional statement p→q. In particular, there are three related conditional statements that occur so often that they have special names.

The proposition q→p is called the converse of p→q. The contrapositive of p→q is the proposition ¬q→¬p. The proposition ¬p→¬q is called the inverse of p→q. We will see that of these three conditional statements formed from p→q, only the contrapositive always has the same truth value as p→q

Which is the correct English translation of ∀x∃y(A(y) ^ (S(x,y) v Ez(F(x,y) ^ S(z,y)))) where S(x,y) is "X speaks y," A(x) is "x is an Asian language," and F(x,y) is "x is a friend of y." The domain of x and z is all the students in a class and domain of y is all languages. 1. Every student in the class speaks an Asian language and has a friend who speaks an Asian language 2. There is a student in this class who speaks an Asian language and has a friend who speaks an Asian language 3. Every student in the class either speaks an Asian language or has a friend who speaks an Asian language 4. All students in the class either speaks an Asian language or has a friend who speaks an Asian language

3. Every student in the class either speaks an Asian language or has a friend who speaks an Asian language 4. All students in the class either speaks an Asian language or has a friend who speaks an Asian language

Which are propositions? 1. Do you still have my copy of Moby dick? 2. x * y = z +4 3. I have read Moby Dick 3 times 4. Put all the chess pieces on the table 5. All the chess pieces are on the table 6. Jessica owns a car

3. I have read Moby Dick 3 times 5. All the chess pieces are on the table 6. Jessica owns a car

Which of these represents ∃xP(x), where P(x) is the statement that x has more than 1 GB RAM and the domain is all the computers at a university? 1. There are at least 1000 computers at the university with more than 1 GB RAM 2. All the computers at the university have more than 1 GB RAM 3. Some of the computers at the university have more than 1 GB RAM 4. There is a computer at the university that has more than 1 GB RAM

3. Some of the computers at the university have more than 1 GB RAM 4. There is a computer at the university that has more than 1 GB RAM

What do these sentences have in common? "George Boole was either born in England or Ireland" and "Carol is either in the attic or the garage"

They use Or with an exclusive Meaning. George could not be born in both places, and Carol could not be in two places at once.

Express in propositional logic "The strike will last less than 100 days only if either the negotiation succeeds and the president of the firm does not resign, or the governor calls out the national guard. " The propositions used are s: "The strike lasts 100 days or more," n: "the negotiation succeeds," and g: "the governor calls out the national guard."

¬s → ((n ^ ¬r) v g)

Let p be "Alex worked for at least 10 hours on the assignment," and q be "Sruthi worked for less than 5 hours." Then the sentence "Alex worked for at least 10 hours on the assignment, but Sruthi worked for less than 5 hours." is represented as p ^ q. What is a correct way to express the negation of this compound proposition?

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

Match the English sentence with the correct Statement in propositional logic, where the proposition p is "The Student passed the exam," and q is "The Student has read the book." It is either "p→q," "q→p," "p^q," or "None of these."

"If the Student pass the exam, then the student has read the Book." is p→q "The student has passed the exam if the student has read the book." is q→p "The student has passed the exam or read the book." is None of these. "The student has passed the exam and has read the book." is p ^ q

What makes these true? 1. ∃x(x>5) 2. ∃x(x=< 0) 3. ∃x(x^2 =1)

1. {2,4,7} The number 7 satisfies it 2. {-5,5} The number -5 satisfies it 3. {1,2,3,4,5} The number 1 satisfies it

1. p→q 2. p V q 3. ¬(p→q) 4. p^ q 5. (p→q) ^ (p→r) 6. p ↔q

1. ¬p V q 2. ¬ p→ q 3. p^ ¬q 4. ¬(p→ ¬q) 6. (p ^q) V (¬p ^ ¬q) 5. p →(q ^ r)

For which of the following is ∀x∀y∃zQ(x,y,z) true? Assuming the domain of quantification for all variables is real numbers. 1. Let Q(x,y,z) be the statement "(xy) / z = x / (yz)" 2. Let Q(x,y,z) be the statement "xy =z" 3. Let Q(x,y,z) be the statement "x / y =z" 4. Let Q(x,y,z) be the statement "(xy)z=x(yz)"

2. Let Q(x,y,z) be the statement "xy =z" 4. Let Q(x,y,z) be the statement "(xy)z=x(yz)"

Premises "If the weather is good, people go to the beach," "If people go to the beach, they are happy," and "The weather is good" lead to the conclusion "People are happy." Let g be the proposition "the weather is good," Let b be the proposition "people go to the beach," Let h be the proposition "People are happy." Which of these is a Modus ponens using a premise and an intermediary result? 1. g 2. b 3. h 4. g → b 5. b → h

3. h

1. (p^ ¬q) ^r means 2. ¬(p v q) v r means 3. ¬p v (q v r) means 4. p ^ (¬q v r) means

3. ¬((p^ ¬q) ^ ¬r) 1. ¬(¬p V q) ^ r 2. r V (¬p ^ ¬q) 4. (p ^ r) V (p ^ ¬q)

Besides "for all" and "for every," universal quantification can be expressed in many other ways, including "all of," "for each," "given any," "for arbitrary," "for each," and "for any."

It is best to avoid using "for any x" because it is often ambiguous as to whether "any" means "every" or "some." In some cases, "any" is unambiguous, such as when it is used in negatives: "There is not any reason to avoid studying."

What do these sentences have in common? "All entrees come with mashed potatoes or steamed vegetables." "You may eat dinner this evening in the cafeteria or in the dining hall."

They use Or with an exclusive Meaning. The restaurant does not provide both without an extra charge, and you cannot eat in both simultaneously.

What do these sentences have in common? "You cannot go on the ride if you are under 4 feet tall or if you weigh over 300lbs." "To enjoy winter, you have to know how to ski or how to skate."

They use Or with an inclusive Meaning. It is possible to be over 300lbs and under 4ft, and it is possible to know how to both ski and skate and enjoy winter.