Sample Questions Exam 1 225

Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer.

(-1)^2m = (-1)^2^m = 1^m

In the expressions below, n is an integer. Indicate whether each expression has a value that is an odd integer or an even integer. Use the definitions of even and odd to justify your answer. You can assume that the sum, difference, or product of two integers is also an integer. -2n^2 - 5

-2n^2 (even) - 2k+1 (odd) = odd

Show that -73 is odd by giving an integer k such that -73 = 2k+1.

-73-1 = -74 /2 = -34

In the expressions below, n is an integer. Indicate whether each expression has a value that is an odd integer or an even integer. Use the definitions of even and odd to justify your answer. You can assume that the sum, difference, or product of two integers is also an integer. 10n^3 + 8n - 4

10n^3 + 8n - 4 is even. 10n^3 + 8n - 4 = 2(5n^3 + 4n - 2). Since n is an integer, 5n^3 + 4n - 2 is also an integer. Since 10n^3 + 8n - 4 is equal to two times an integer, 10n^3 + 8n - 4 is even.

Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. x is an even integer then (-1)^x = -1

2(2k^2 + 2k) + 1

Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. The sum of an odd and an even integer is odd.

2(k+j) + 1

About Find a counterexample to show that each of the statements is false. For every positive integer x, x^3 < 2^x.

Any positive integer that is greater than 1 and less than 10 is a counterexample.

Consider the following pieces of identification a person might have in order to apply for a credit card: B: Applicant presents a birth certificate. D: Applicant presents a driver's license. M: Applicant presents a marriage license. Write a logical expression for the requirements under the following conditions: The applicant must present either a birth certificate, a driver's license or a marriage license.

BvDvM

Consider the following pieces of identification a person might have in order to apply for a credit card: B: Applicant presents a birth certificate. D: Applicant presents a driver's license. M: Applicant presents a marriage license. Write a logical expression for the requirements under the following conditions: Applicant must present either a birth certificate or both a driver's license and a marriage license.

BvD∧M

Consider the following pieces of identification a person might have in order to apply for a credit card: B: Applicant presents a birth certificate. D: Applicant presents a driver's license. M: Applicant presents a marriage license. Write a logical expression for the requirements under the following conditions: The applicant must present at least two of the following forms of identification: birth certificate, driver's license, marriage license.

B∧MvB∧DvD∧M

Have a nice day.

Command, not a proposition.

Use the laws of propositional logic to prove the following: (p ∧ q) → r ≡ (p ∧ ¬r) → ¬q

Conditional demorgan's associative commutative associative double negation demorgan's law conditional

Use the laws of propositional logic to prove the following: ¬p → ¬q ≡ q → p

Conditional double negation commutative conditional

Show whether each logical expression is a tautology, contradiction or neither. (¬p ∨ q) ↔ (p ∧ ¬q)

Contradiction-every value is false

Even or odd? n = 258

Even

Which of the following conditional statements are true and why? If January has 31 days, then 7 is an even number.

False. The hypothesis is true and conclusion is false.

Determine whether each statement is true or false. Provide a justification for each answer. Showing that a statement holds for a few cases is sufficient to prove a universal statement.

False. To show that a universally quantified statement ∀x∈D,P(x) is true, the predicate P(x) must hold for all x values in the domain D. For universal statements with an infinite domain, proving that the predicate holds for a few cases is insufficient. Ex: For all integers x, x2 is positive. Although (−1)2 and 12=1 are both positive, many other integers exist. The statement needs to be proved for all integers, not just −1 and 1.

For each statement below state what needs to be proven in order to show that the existential statement is false. Your response should be a universal statement. There exists a negative integer that is equal to its square.

For every negative integer x, x ≠ x2.

Express the following compound propositions in English using the following definitions: p: I am going to a movie tonight. q: I am going to the party tonight. ¬(p ∧ q)

I am not going to a party and a movie tonight

Express the following compound propositions in English using the following definitions: p: I am going to a movie tonight. q: I am going to the party tonight. ¬p ∨ ¬q

I am not going to a party or I am not going to a movie

Write each of the following statements as a precise mathematical statement. Use variable names to denote arbitrary numbers in the domain. Your statements should avoid mathematical terms in text (such as "square root" or "multiplicative inverse") and should be expressed using algebra. The square root of every positive number less than one is greater than the number itself.

If x is a real number and 0 < x < 1, then x>x.

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. January has exactly 31 days or April has exactly 30 days.

Inclusive or: True Exclusive or: False

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. There are eight days in a week or there are seven days in a week.

Inclusive or: True Exclusive or: True

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. The number π is an integer or the sun revolves around the earth.

Inclusive or: false Exclusive or: false

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. 20 nickels are worth one dollar or whales are mammals.

Inclusive or: true Exclusive or: false

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. February has 31 days or the number 5 is an integer.

Inclusive or: true Exclusive or: true

State the inverse, contrapositive, and converse of each conditional statement. Then indicate whether the inverse, contrapositive, and converse are true. If 3 is a prime number then 5 is an even number.

Inverse: If 3 is not a prime number, then 5 is not an even number. (FT True) Contrapositive: If five is not an even number, then 3 is not a prime number. (TF False) Converse: If 5 is an even number, then 3 is a prime number. (FT True)

Give the inverse, contrapositive, and converse for each of the following statements: If it snowed last night, then school will be cancelled

Inverse: If it did not snow last night, then school will not be cancelled. Contrapositive: If school will not be cancelled, then it did not snow last night. Converse: If school will be cancelled, then it snowed last night.

Give the inverse, contrapositive, and converse for each of the following statements: If she finished her homework, then she went to the party.

Inverse: If she did not finish her homework, she did not go to the party. Contrapositive: if she did not go to the party, then she did not finish her homework. Converse: If she went to the party, then she finished her homework.

Give the inverse, contrapositive, and converse for each of the following statements: If the patient took the medicine, then she had side effects.

Inverse: If the patient did not take the medicine, then she did not have side effects. Contrapositive: If she did not have side effects, then she did not take the medicine. Converse: If she had side effects, then she took the medicine.

Give the inverse, contrapositive, and converse for each of the following statements: If he trained for the race, then he finished the race.

Inverse:If he did not train for the race, then he did not finish the race. Contrapositive: If he did not finish the race, then he did not train for the race. Converse: If he trained for the race, then he finished the race.

Which of the following are functions from R to R? If f is a function, give its range. f(x) = sqrt(x^2)

Is a function because it is always positive due to x^2.

Write each of the following statements as a precise mathematical statement. Use variable names to denote arbitrary numbers in the domain. Your statements should avoid mathematical terms in text (such as "square root" or "multiplicative inverse") and should be expressed using algebra. Every real number besides 0 has a multiplicative inverse.

Let x be a real number such that x ≠ 0, then there is a real number y such that xy = 1.

Which of the following are functions from R to R? If f is a function, give its range. f(x) = 1/(x^2-4)

Not a function. If x = 2 then the number blows up to infinity.

perfect square

One of the questions below is about a perfect square. A number n is a perfect square if n = k2 for some integer k.

2 + 3 = 6

Proposition. Negation: 2+3!=6

It's a beautiful day.

Proposition. Negation: It is not a beautiful day

The patient has diabetes.

Proposition. Negation: The patient does not have diabetes.

Every prime number is even.

Proposition. Negation: every prime number is not even.

The light is on.

Proposition. Negation: the light is not on

The sky is purple.

Proposition. Negation: the sky is not purple.

The soup is cold.

Proposition. Negation: the soup is not cold

There is a number that is larger than 17.

Proposition. Negation: there is not a number larger than 17.

Do you like my new shoes?

Question, not a proposition

Determine whether the following propositions are true or false: 5 is an odd number and 3 is a negative number.

Since 3 is not a negative number, the proposition is false.

Determine whether the following propositions are true or false: 5 is an odd number or 3 is a negative number.

Since 5 is an odd number, the proposition is true.

Determine whether the following propositions are true or false: 6 is an even number and 7 is odd or negative.

Since 6 is an even number and 7 is an odd number, the statement is true.

Determine whether the following propositions are true or false: It is not true that either 7 is an odd number or 8 is an even number (or both).

Since 7 is an odd number, or is true but its negation is false.

Determine whether the following propositions are true or false: 8 is an odd number or 4 is not an odd number.

Since neither of these propositions are true, the statement is false.

Axioms

Statements assumed to be true

Show whether each logical expression is a tautology, contradiction or neither. (p ∨ q) ∨ (q → p)

Tautology-every value is true

Suppose that p, q, r, s, and t are all propositional variables. Describe in words when the expression p ∧ q ∧ r ∧ s ∧ t is true and when it is false.

The expression is true when all of the variables are true and false when at least one of the variables is false.

Suppose that p, q, r, s, and t are all propositional variables. Describe in words when the expression p ∨ q ∨ r ∨ s ∨ t is true and when it is false.

The expression is true when at least one of the variables is true and false when all of the propositional variables are false

Prove that the following pairs of expressions are not logically equivalent. p ∧ (p → q) and p ∧ q

They are equivalent

Use the definitions for the sets given below to determine whether each statement is true or false: A = { x ∈ Z: x is an integer multiple of 3 } B = { x ∈ Z: x is a perfect square } C = { 4, 5, 9, 10 } D = { 2, 4, 11, 14 } E = { 3, 6, 9 } F = { 4, 6, 16 } An integer x is a perfect square if there is an integer y such that x = y2. 27 ∈ A

True. 3*9 = 27

Determine whether each statement is true or false. Provide a justification for each answer. Proof by exhaustion can be used to prove a universally quantified statement with a finite domain.

True. If the domain of a universal statement is finite, then the statement can be checked for each value in the domain.

Which of the following conditional statements are true and why? If February has 30 days, then 7 is an odd number.

True. The hypothesis is false and the conclusion is true.

Which of the following conditional statements are true and why? If 7 is an odd number, then February does not have 30 days.

True. The hypothesis is true and conclusion is true.

Theorem

a statement that can be proven to be true

prime? composite? neither? n = 21

composite: 3*7

Use the laws of propositional logic to prove that each statement is a tautology. (p ∧ q) → (p ∨ r)

conditional demorgan's associative commutative associative comutative complement commutative domination commutative domination

Use the laws of propositional logic to prove that each statement is a tautology. ¬r ∨ (¬r → p)

conditional double negation associative commutative complement commutative domination

Prove that the following pairs of expressions are not logically equivalent. ¬p → q and ¬p ∨ q

create the truth table Different when both were false and p is true and q is false

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? p ∨ r

false

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? q ⊕ s

false

Define the following propositions: j: Sally got the job. l: Sally was late for her interview r: Sally updated her resume. Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent. If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and was not late for her interview, then she got the job.

logically equivalent

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient had nausea or migraines.

n ∨ m

prime? composite? neither? n = 1

neither. It is not prime because it is not above 1.

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient had nausea and migraines.

n∧m

Indicate whether each integer n is even or odd. If n is even, show that n equals 2k, for some integer k. If n is odd, show that n equals 2k+1, for some integer k. n = -1

odd

Give truth values for the propositional variables that cause the two expressions to have different truth values. For example, given p ∨ q and p ⊕ q, the correct answer would be p = q = T, because when p and q are both true, p ∨ q is true but p ⊕ q is false. Note that there may be more than one correct answer. r ∧ (p ∨ q) (r ∧ p) ∨ q

p = T, q = T, r = F or p = F, q = T, r = F.

Give truth values for the propositional variables that cause the two expressions to have different truth values. For example, given p ∨ q and p ⊕ q, the correct answer would be p = q = T, because when p and q are both true, p ∨ q is true but p ⊕ q is false. Note that there may be more than one correct answer. p ∨ q (¬p ∧ q) ∨ (p ∧ ¬q)

p and q are true or p and true are false

Prove each existential statement given below. There are integers m and n such that sqrt(m+n)=sqrt(m)+sqrt(n)

sqrt of 0 and 4

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? q ∧ s

true

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? q ∨ s

true

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? q ⊕ r

true

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? ¬p

true

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient took the medication, but still had migraines.

t∧m

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. Despite the fact that the patient took the medication, the patient had nausea.

t∧n

Prove each existential statement given below. There are positive integers x and y such that 1/x+1/y is an integer.

x and y equal 2

Rational? n = pi/6pi

x/y x= pi y = 6 pi

Show that each number n is rational by showing that n is equal to the ratio of two integers, where the denominator is non-zero. n = .3274

x/y x=3274 y = 10000!= 0

Give an equivalent statement for each statement. Your answer should be one of the following: It is not true that x > 7

x<=7

Express the range of each function using roster notation. Let A = {2, 3, 4, 5}.f: A → Z such that f(x) = 2x - 1.

{3,5,7,9}

Give a logical expression with variables p, q, and r that is true if p and q are false and r is true and is otherwise false.

~(p∧q)∧r

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient did not have migraines.

~m

Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English. p: the applicant has written permission from his parents e: the applicant is at least 18 years old s: the applicant is at least 16 years old

~pv~s

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. There is no way that the patient took the medication.

~t