Discrete Mathematics

True or False: The statement ~ p V q is a contradiction

False

n is prime n > 1

n=rs, if and only if r and s are positive integers, then r or s equals n

What is a tautology?

A statement that is always true by logical structure (I believe..., I think...)

Write the statements in symbolic form using the symbols ~, ⋁, and ⋀ and the indicated letters to represent component statements. Let h = "John is healthy," w = "John is wealthy," and s = "John is wise." John is healthy and wealthy but not wise.

(h ⋀ w) ⋀ ~s

Let p, q, and r be statements. Completely distribute the following negations (your answer should only contain the following symbols: parenthesis, ~, ∧, and ∨): ~(p↔q) : hint p↔q ≡ (p→q) ∧ (q→p)

(p ∧ ~q) ∨ (q ∧ ~p)

Let p, q, and r be statements. Fully distribute the following negations: ~((p ∧ ~q) ∨ ~(q ∧ ~r))

(~p ∨ q) ∧ (q ∧ ~r)

The following statement is false. Give a counterexample. For every integer n, if n is prime then (−1)^n = −1. n =

2

Use the unique factorization theorem to write the following integers in standard factored form. (a) 2,646

2^1 * 3^3 * 7^2

Now prove the negation. Suppose n is any integer. Express 6n^2 + 27 as the following product: 6n^2 + 27 = 3(___________)

2n^2+9

"If p then q" What is p?

hypothesis

Let m = "Haley is a math major" d= "Haley is taking Discrete Mathematics" c = "Haley is taking Calculus I" Rewrite the following statements in symbolic form: (a) "Haley is a math major and taking Discrete Math."

m ∧ d

Let p be a statement, t be a tautological statement, and c be a contradictory statement. p ∧ t ≡

p

Let p be a statement, t be a tautological statement, and c be a contradictory statement. p ∨ c ≡

p

Converse Error

p then q q therefore p

Let p, q, and r be statements. Fully distribute the following negations: ~(~p ∨ q)

p ∧ ~q

What is logically equivalent to ~(p→q)

p ∧ ~q

What is the converse of p→q

q→p

Let p be a statement, t be a tautological statement, and c be a contradictory statement. p ∨ t ≡

t

Rewrite each statement without using variables or the symbol ∀ or ∃. Indicate whether the statement is true or false. (a) ∀ nonzero real number r, ∃ a real number s such that rs = 1. Is the statement true or false?

true

Find the truth set of the following predicate within the given domain: Predicate: -2 ≤ d ≤ 2. Domain: ℤ

{-2, -1, 0, 1, 2}

Find the truth set of the following predicate within the given domain: (b) Predicate 12/d is an integer. Domain: ℤ

{1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12}

Find the truth set of the following predicate within the given domain: (a) Predicate: 12/d is an integer. Domain: ℕ (=ℤ+)

{1, 2, 3, 4, 6, 12}

Find the truth set of the following predicate within the given domain: Predicate: -2 ≤ d ≤ 2. Domain: ℕ (=ℤ+)

{1, 2}

contradiction rule

~p → c ∴ p

What is the negation of p ∧ (q ∨ r)

~p ∨ (~q ∧ ~r)

De Morgan's Law ~ (p ∧ q) ≡

~p ∨ ~q

Write a formal negation for each of the following statements: ∀y ∈ ℝ, y^(1/2) ≥ 0, y ≥ 0

∃y ∈ ℝ such that y^(1/2) < 0 and y < 0

Consider the following. ∀r ∈ Q, ∃ integers a and b such that r = a/b. Rewrite the statement in English without using symbols or variables and express your answer as simply as possible.

Every rational number is equal to a ratio of some two integers.

Let P(x) be the predicate " x^3 > 0" with domain ℝ. Determine wheter the following statements are true or false: True or False: P(-1)

False

Let P(x) be the predicate " x^3 > 0" with domain ℝ. Determine wheter the following statements are true or false: True or False: ∀x ∈ ℝ, ~P(x)

False

Let Q(x) be the predicate "If x < 2, then x^2 < 4" with domain ℝ. Determine the following statements are true or false: True or False: Q(-3)

False

Let Q(x) be the predicate "If x < 2, then x^2 < 4" with domain ℝ. Determine the following statements are true or false: True or False: ∀x ∈ ℝ, ~Q(x)

False

True or False: The statement ~ p V q is a tautology

False

Modus Tollens

If P then Q Not Q Therefore not P

Definition: If r and s are statements, r unless s means if ~s then r. Rewrite the following statement in if-then form. This door will not open unless a security code is entered.

If a security code is not entered, then the door will not open.

Let p = "My car runs" and q = "My car has an engine." What is the contrapositive: ~q ⇒ ~p

If my car does not have an engine, then my car does not run

What makes an argument Valid.

If the premise are all true and the resulting conclusion is true

Rewrite the following statement as a conjunction of two if-then statements. This integer is even if, and only if, it equals twice some integer.

If this integer is even, then it equals twice some integer, and if this integer equals twice some integer, then it is even.

Consider the statement: "If today is New Year's Eve, then tomorrow is January." (b) What is its inverse?

If today is not NYE, then tomorrow is not January.

Consider the statement: "If today is New Year's Eve, then tomorrow is January." (d) What is its contrapositive?

If tomrrow is not January, then today is not NYE.

Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) If x is nonnegative, then x is positive or x is 0. (d) Inverse

If x is not nonnegative, then both x is not positive and x is not 0.

Rewrite the following statements in "if-then" form: (b) Eat dinner or you won't get dessert.

If you don't eat dinner, then you won't get dessert

Rewrite the following statements in "if-then" form: (a) Fix my ceiling or I won't pay my rent

If you don't fix my ceiling, then I won't pay my rent.

Negate "If my car is in the repair shop, then I cannot get to class"

My car is in the repair shop and I can get to class

What are statments before the conclusion called?

Premise

Write a negation for each of the following statements: All cats are orange

Some cats are not orange

Which of the following is a negation for "All dogs are loyal"? (Select all that apply.)

Some dogs are disloyal; There is a dog that is disloyal.

What is the conclusion

The final statement in an argument

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (c)∀x in D, ∃y in E such that xy ≥ y. Negation: ∃x in D such that ∀y in E, xy < y. Which is true: ( the given statement or the negation)

The given statement

Consider the statement: "If today is New Year's Eve, then tomorrow is January." (a) What is its negation?

Today is NYE and tomorrow is not January.

(a) ∃x such that Rect(x) ∧ Square(x) This statement is (T or F)

True

Let P(x) be the predicate " x^3 > 0" with domain ℝ. Determine wheter the following statements are true or false: True or False: P(1)

True

Let P(x) be the predicate " x^3 > 0" with domain ℝ. Determine wheter the following statements are true or false: True or False: ∃x ∈ ℝ such that P(x)

True

Let Q(x) be the predicate "If x < 2, then x^2 < 4" with domain ℝ. Determine the following statements are true or false: True or False: Q(1)

True

Let Q(x) be the predicate "If x < 2, then x^2 < 4" with domain ℝ. Determine the following statements are true or false: True or False: ∃x ∈ ℝ such that Q(x)

True

What is a contradiction?

Two statements that have opposite meanings

Consider the following argument: If 1032 is divisible by 6, then it is divisible by 3. 1032 is divisible by 6 Therefore 1032 Choose the one that accurately describes the argument form: a. Valid, by modus ponens. b. Valid, by modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

a

conditional statement

a statement that can be written in if-then form

biconditional statement

a statement that can be written in the form "p if and only if q". Another form is p ⇔ q ≡ (p ⇒ q) ∧ (q ⇒ p)

Fallacy

an error in reasoning that results in an invalid argument

Consider the statment: ∃x ∈ D such that x^2 > 1 Which of the following are equivalent ways of expressing this statement? a. If x is positive, negative, or zero, then it is a real number b. Each real number is positive, negative, or zero. c. No real numbers are neither positive, negative, nor zero. d. x is positive, negative, or zero, for any real number x. e. Some real numbers are positive, negative, or zero. f. Given a real number that is positive, negative or zero, it is x.

b, c, d

Consider the following argument: If my car runs, then it has gas. My car has gas. Therefore My car runs. Choose one the accurately describes the argument form: a. Valid, by modus ponens. b. Valid, by modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

c

Let p be a statement, t be a tautological statement, and c be a contradictory statement. p ∧ c ≡

c

a ≤ x ≤ b is logically equivallent to?

(a ≤ x) ∧(x ≤ b)

Determine whether the statements in A and B are logically equivalent. Statement A: Bob is both a math and computer science major and Ann is a math major, but Ann is not both a math and computer science major. Statement B: It is not the case that both Bob and Ann are both math and computer science majors, but it is the case that Ann is a math major and Bob is both a math and computer science major. Let b be "Bob is a double math and computer science major," m be "Ann is a math major," and a be "Ann is a double math and computer science major." Create a truth table for the two statements. Write the statements in symbolic form using the symbols ~, ∨, and ∧ and the indicated letters to represent component statements. Statement A

(b ∧ m) ∧ ~a

The following statement is true: "∀ nonzero number x, ∃ a real number y such that xy = 1." For each x given below, find a y to make the predicate "xy = 1" true. (a) x = 5 Let y =

1/5

The following statement is true: "∀ real number x, ∃ an integer n such that n > x." For each x given below, find an n to make the predicate "n > x" true. (b) x = 104 Let n =

10^5

The following statement is true: "∀ real number x, ∃ an integer n such that n > x." For each x given below, find an n to make the predicate "n > x" true. (a) x = 12.83 Let n =

13

Disprove the following statement by giving a counterexample. (Enter your answers as a comma-separated list.) For every integer p, if p is prime then p^2 − 1 is even. (p, p^2 − 1) =

2,3

Use the unique factorization theorem to write the following integers in standard factored form. (b) 4,851

3^2 * 7^2 * 11^1

The following statement is true: "∀ nonzero number x, ∃ a real number y such that xy = 1." For each x given below, find a y to make the predicate "xy = 1" true. (c) x = 4/5 Let y =

5/4

The following statement is true: "∀ real number x, ∃ an integer n such that n > x." For each x given below, find an n to make the predicate "n > x" true. (c)x = 5^5^5 Let n =

5^5^6

Determine if the following statement is true or false. (Enter TRUE if the statement is true, enter a counterexample if the statement is false). If m and n are any positive integers and mn is a perfect square, then m and n are perfect squares. (m, n) =

6,6

Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous. (d) Some estimates are accurate.

All estimates are inaccurate.

Write a negation for each of the following statements: Some movies are too short or too long

All movies are not too short and not too long

Consider the statement "There are no simple solutions to life's problems." Write the informal negation of the statement.

At least one of life's problems has a simple solution

Rewrite each statement without using quantifiers or variables. Indicate which are true and which are false. Let the domain of x be the set of geometric figures in the plane, and let Square(x) be "x is a square" and Rect(x) be "x is a rectangle." (c) ∀x, Square(x) → Rect(x)

For all geometric figures, if the geometric figure is a square then it is also a rectangle.

Write a negation for each of the following statements: Given some triangle, if its three sides are equal, then it is equilateral.

For any triangle, its three sides are equal and it is not equilateral.

Prove that the following statement is false. There exists an integer n such that 6n2 + 27 is prime. To prove the statement is false, prove the negation is true. Write the negation of the statement.

For every integer n, 6n^2 + 27 is not prime.

Rewrite each statement without using variables or the symbol ∀ or ∃. Indicate whether the statement is true or false. (a) ∀ nonzero real number r, ∃ a real number s such that rs = 1.

Given any nonzero real number, a real number can be found so that the product of the two equals 1.

Let p = "My car runs" and q = "My car has an engine." What is the inverse: ~p ⇒ ~q

If my car does not run, then my car does not have an engine

Let p = "My car runs" and q = "My car has an engine." What is the converse q ⇒ p

If my car has an engine, then my car runs

Let p = "My car runs" and q = "My car has an engine." What is p ⇒ q?

If my car runs, then my car has an engine

The zero product property, says that if a product of two real numbers is 0, then one of the numbers must be 0. (c) Write an informal version (without quantifier symbols or variables) for the contrapositive of the zero product property.

If neither of two real numbers is zero, then their product is nonzero.

Consider the statement: "If today is New Year's Eve, then tomorrow is January." (c) What is its converse?

If tomorrow is January, then today is NYE.

Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) If x is nonnegative, then x is positive or x is 0. (b) Contrapositive

If x is not positive and x is not 0, then x is not nonnegative.

Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) If x is nonnegative, then x is positive or x is 0. (c) Converse

If x is positive or x is 0, then x is nonnegative.

Rewrite the statement in if-then form. Fix my ceiling or I won't pay my rent.

If you don't fix my ceiling, then I won't pay my rent.

Negate " It is Sunday and it is raining."

It is not Sunday or it is not raining

The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You are visiting the island and have the following encounters with natives. (c) You then encounter natives E and F. E says: F is a knave. F says: E is a knave. How many knaves are there?

One, only one of them is a knave.

Prove: If r and s are integers, then there exists an integer such that 22r + 18s = 2k

Proof: Assume that r and s are integers, and let k=11r + 9s. K is an integer and 2k=22r + 18s. Since 11, 9 , r and s are all integers. 11r + 9s is an integer and therefore, k is an integer. Now, 2k=2(11r + 9s) = 22r + 18s. Hence, there exists an integer, namely k such that 22r + 18s = 2k.

Write a negation for each of the following statements: Every party is fun

Some parties aren't fun

The following statement is true. (Note that you do not need to understand the statement in order to be able to do this exercise.) For every real number x, if x > 1 then x^2 > x. (b) Write the first sentence of a proof (the "starting point").

Suppose x is any real number such that x > 1.

Determine whether the argument forms is valid and explain how the truth table supports your answer.

The argument is invalid because there exists a row that has true premises and a false conclusion.

Determine whether the proposed negation is correct. If it is not, write a correct negation. Statement: The product of any irrational number and any rational number is irrational. Proposed negation: The product of any irrational number and any rational number is rational.

The proposed negation is not correct. A possible correct negation would be: There is an irrational number and a rational number whose product is rational.

Consider the following. ∀r ∈ Q, ∃ integers a and b such that r = a/b. Write a negation for the statement.

There is at least one rational number that is not equal to a ratio of any two integers.

The following statement is true. (Note that you do not need to understand the statement in order to be able to do this exercise.) For every real number x, if x > 1 then x^2 > x. Write the last sentence of a proof (the "conclusion to be shown").

Therefore x^2 > x.

True or False: The Statements p→q and ~p ∨ q are logically equivalent

True

What does it mean when two statements are logically equivalent?

When both statments have the same meanings.

Vacuous Truth

When the hypothesis is false, the conclusion is true.

Justify your answers by using the definitions of even, odd, prime, and composite numbers. Assume that c is a particular integer. (c) Is (c^2 + 1) − (c^2 − 1) − 2 an even integer?

Yes, because (c^2 + 1) − (c^2 − 1) − 2 = 2(0) and 0 is an integer.

Give a reason for your answer to the following question. Does '3'|'0' ?

Yes, because 0 = 3 · 0.

Give a reason for your answer to the following question. Is 48 divisible by 12?

Yes, because 48 = 12 · 4.

Justify your answer by using the definitions of even, odd, prime, and composite numbers. Assume that r and s are particular integers. (a) Is 4rs even?

Yes, because 4rs = 2(2rs) and 2rs is an integer.

Justify your answers by using the definitions of even, odd, prime, and composite numbers. Assume that c is a particular integer. (b) Is 6c + 3 an odd integer?

Yes, because 6c + 3 = 2(3c + 1) + 1 and 3c + 1 is an integer.

Justify your answer by using the definitions of even, odd, prime, and composite numbers. Assume that r and s are particular integers. (b) Is 6r + 8s^2 + 5 odd?

Yes, because 6r + 8s^2 + 5 = 2(3r + 4s^2 + 2) + 1 and 3r + 4s^2 + 2 is an integer.

Give a reason for your answer to the following question. Assume that all variables represent integers. If n = 8k + 1, does 16 divide n2 − 1?

Yes, because n2 − 1 = (8k + 1)^2 − 1 = (64k^2 + 16k + 1) − 1 = 16(4k^2 + k) which is a product of integers because the sums and products of integers are integers.

Justify your answer by using the definitions of even, odd, prime, and composite numbers. Assume that r and s are particular integers. (c) If r and s are both positive, is r2 + 2rs + s2 composite?

Yes, because r^2 + 2rs + s^2 = (r + s)^2 and r + s is an integer.

Justify your answers by using the definitions of even, odd, prime, and composite numbers. Assume that c is a particular integer. (a) Is −4c an even integer?

Yes, because −4c = 2(−2c) and −2c is an integer.

Consider the statement: ∀n ∈ ℤ, if n is even, n^2 is even Which of the following are equivalent ways of expressing this statement? a. All squares of even integers are even b. All integeres are even and have even squares c. Any integer that is even has an even square. d. Given any integer that is even, that integers square is even. e. For all integers, there are some whose square is even. f. If an integer is even, then that integer squared is even.

a, c, d, f

The conjunction "but" has the same logical meaning as?

and

Consider the following argument -2 ∈ ℕ ⇒ -2 > 0 -2 ~> 0 -2 ∉ ℕ Choose the one that accurately describes the argument form: a. Valid, by modus ponens. b. Valid, by modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

b

Consider the arument: For any x,y ∈ ℝ, if x >0 and y > 0 then xy > 0 The numbers a and b are particular real numbers such that ab > 0 Therefore, a >0 and b > 0 Choose the one the accurately describes the argument form: a. Valid by univeral modus ponens. b. Valid, by universal modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

c

Consider the following argument: If I complete this study guide, then I will do well on the test. I did not complete this study guide. Therefore I will not do well on the test. Choose the one that accurately describes the argument form: a. Valid, by modus ponens. b. Valid, by modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

d

Consider the following theorem. Theorem: The sum of any even integer and any odd integer is odd. Six of the sentences in the following scrambled list can be used to prove the theorem. (a) Let t = r + s. Then, t is an integer because it is a sum of integers. (b) Hence, the sum is twice an integer plus one. So by definition of odd, m + n is odd. (c) By definition of even and odd, there are integers r and s such that m = 2r and n = 2s + 1. (d) So by definition of even, t is even. (e) By definition of even and odd, there is an integer r such that m = 2r and n = 2r + 1. (f)Suppose m is any even integer and n is any odd integer. (g) Let m + n be any odd integer. (h) By substitution and algebra, m + n = 2r + (2s + 1) = 2(r + s) + 1. (i) By substitution, m + n = 2t + 1.

f, c, h, a, i, b

Rewrite each statement without using variables or the symbol ∀ or ∃. Indicate whether the statement is true or false. (b) ∃ a real number r such that ∀ nonzero real numbers s, rs = 1. Is the statement true or false?

false

The argument below may be valid or exhibit the converse or the inverse error. Use symbols to write the logical form of the argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If there are as many rational numbers as there are irrational numbers, then the set of all irrational numbers is infinite. The set of all irrational numbers is infinite ∴ There are as many rational numbers as there are irrational numbers. Let p = "there are as many rational numbers as there are irrational numbers" and q = "the set of all irrational numbers is infinite."

form: p → q invalid, converse error q ∴ p

Consider the following statement. If 1 − 0.99999... is less than every positive real number, then it equals zero. Use modus ponens or modus tollens to fill in the blanks in the argument so as to produce a valid inference. 1 − 0.99999... is ________________________ real number. ∴ The number 1 − 0.99999... equals zero.

less than every positive

Let m = "Haley is a math major" d= "Haley is taking Discrete Mathematics" c = "Haley is taking Calculus I" Rewrite the following statements in symbolic form: (c) " If Haley is a math major, but she is taking neither Discrete Math nor Calculus I."

m ∧ ~(d ∨ c)

Juan is a math major but not a computer science major. (m = "Juan is a math major;" c = "Juan is a computer science major.")

m ⋀ ~c

even integer n

n = 2k for some integer k

n is composite n > 1

n=rs, if and only if there exist integers r and s both between 1 and n (non-inclusive).

Is 1 prime?

no (by definition, the integer must be bigger than 1)

Inverse Error

p then q not p therefore not q

Indicate which column(s) represent the premise(s). (Select all that apply.)

p → q ∨ r ~q ∨ ~r

Let p, q, and r be statements. Completely distribute the following negations (your answer should only contain the following symbols: parenthesis, ~, ∧, and ∨): ~(p→(q→r))

p ∧ (q ∧ ~r)

~ (p→~q) ≡

p ∧ q

Write each of the three statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explanation to show that you understand what it means for statements to be logically equivalent. Let p represent "It walks like a duck," q represent "It talks like a duck," and r represent "It is a duck." (a) If it walks like a duck and it talks like a duck, then it is a duck.

p ∧ q → r

What is the negation of p→q

p ∧ ~q

Write the statement in symbolic form using the symbols ~, ∨, and ∧ and the indicated letters to represent component statements. Let p = "x > 6," q = "x = 6," and r = "11 > x. x ≥ 6

p ∨ q

Now 2n^2+9 is an integer because sums and products of integers are integers. Thus, 6n2 + 27 is not prime because it is a _________ of ____________.

product, two positive integers greater than 1

Write the statement in symbolic form using the symbols ~, ∨, and ∧ and the indicated letters to represent component statements. Let p = "x > 6," q = "x = 6," and r = "11 > x. 11 > x ≥ 6

r ∧ (p ∨ q)

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (a) ∀x in D, ∃y in E such that x + y = 1. Negation: ∃x in D such that ∀y in E, x + y ≠ Which is true

the negation

Write the statements in symbolic form using the symbols ~, ⋁, and ⋀ and the indicated letters to represent component statements. Let h = "John is healthy," w = "John is wealthy," and s = "John is wise." John is wealthy, but he is not both healthy and wise.

w ⋀ ~(h ⋁ s)

The following statement is true. (Note that you do not need to understand the statement in order to be able to do this exercise.) For every real number x, if x > 1 then x^2 > x. (a) Rewrite the statement with the quantification implicit as If _____, then _____.

x is a real number greater than 1, x^2 > x

Let m = "Haley is a math major" d= "Haley is taking Discrete Mathematics" c = "Haley is taking Calculus I" Rewrite the following statements in symbolic form: (e) "If Haley is not taking both Calculus I and Discrete Math, then she is not a Math Major."

~( c ∧ d) → ~m

Let m = "Haley is a math major" d= "Haley is taking Discrete Mathematics" c = "Haley is taking Calculus I" Rewrite the following statements in symbolic form: (b) "Haley is not a math major, but she is taking Calculus I"

~m ∧ c

Let p, q, and r be statements. Completely distribute the following negations (your answer should only contain the following symbols: parenthesis, ~, ∧, and ∨): ~(~p→q)

~p ∧ ~q

Let p, q, and r be statements. Fully distribute the following negations: ~(p ∧ q)

~p ∨ ~q

Consider the following argument: If 41 is not prime, then it is composite. 41 is not composite. therefore 41 is prime or not composite Let p = "41 is prime" and q = "41 is composite." Write the argument form using the symbols ~, ∨, ∧, →, ↔.

~p→q ~q therefore p ∨ ~q

Write the statements in symbolic form using the symbols ~, ⋁, and ⋀ and the indicated letters to represent component statements. Let h = "John is healthy," w = "John is wealthy," and s = "John is wise." John is not wealthy, but he is healthy and wise.

~w ⋀ (h ⋀ s)

Write the statements in symbolic form using the symbols ~, ⋁, and ⋀ and the indicated letters to represent component statements. Let h = "John is healthy," w = "John is wealthy," and s = "John is wise." John is neither healthy, wealthy, nor wise.

~w ⋀ ~h ⋀ ~s

Rewrite each of the following statements in the form "∀ _____ x, _____." (c) No irrational numbers are integers.

∀ irrational numbers x, x is not an integer

Rewrite each of the following statements in the form "∀ _____ x, _____." (d) No logicians are lazy.

∀ logicians x, x is not lazy

Write a formal negation for each of the following statements. (c) ∃ a movie m such that m is over 6 hours long

∀ movie m, m is less than or equal to 6 hours long.

Consider the statement "There are no simple solutions to life's problems." Write the formal version of the original statement.

∀ of life's problems x, x does not have a simple solution

Consider the following st ment. Somebody trusts everybody. (b) Write a negation for the statement.

∀ person x, ∃ a person y such that x does not trust y.

Consider the following statement. Everybody trusts somebody. (a) Rewrite the statement formally using quantifiers and variables.

∀ person x, ∃ a person y such that x trusts y.

The zero product property, says that if a product of two real numbers is 0, then one of the numbers must be 0. (b) Write the contrapositive of the zero product property.

∀ real numbers x and y, if x ≠ 0 and y ≠ 0 then xy ≠ 0.

The zero product property, says that if a product of two real numbers is 0, then one of the numbers must be 0. (a) Write this property formally using quantifiers and variables.

∀ real numbers x and y, if xy = 0 then x = 0 or y = 0.

Rewrite each of the following statements in the form "∀ _____ x, _____." (b) Every real number is positive, negative, or zero.

∀ real numbers x, x is positive, negative, or zero

Rewrite each of the following statements in the form "∀ _____ x, _____." (f) The number −1 is not equal to the square of any real number

∀ real numbers x, x² does not equal −1

Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (b) Every computer science student is an engineering student.

∀ s { D, C(s) → E(s)

Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (c) No computer science students are engineering students

∀ s { D, C(s) → ~E(s)

Let D be the set of all quadrilaterals, and consider the predicates: R(x) : x is rectangle S(x) : x is a square Rewrite the following in symbolic form (that is, with quantifiers, variables, and the predicates above): All quadrilaterals are either rectangles or not squares

∀ x ∈ D, R(x) ∨ ~S(x)

Let D be the set of all quadrilaterals, and consider the predicates: R(x) : x is rectangle S(x) : x is a square Rewrite the following in symbolic form (that is, with quantifiers, variables, and the predicates above): All squares are rectangles.

∀ x ∈ D, S(x)→R(x)

Write a formal negation for each of the following statements: ∃m ∈ ℤ such that m is even and odd

∀m ∈ ℤ, m is not even or not odd

For each of the following, write the negation and determine if the statement or its negation is true: Statement: ∃m ∈ ℤ such that ∀n ∈ ℤ, m^n = 1 Which is true

∀m ∈ ℤ, ∃n ∈ℤ such that m^n≠1, statement

Write a formal negation for each of the following statements: ∃x ∈ ℝ such that x^2 < 0

∀x ∈ ℝ, x^2 ≥ 0 or ∀x ∈ ℝ, x^2 ~≤ 0

For each of the following, write the negation and determine if the statement or its negation is true: Statement: ∃x ∈ ℝ such that ∀y ∈ ℝ, x+y = 0 Which is true?

∀x ∈ ℝ, ∃y ∈ ℝ such that x+y ≠ 0, Negation

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (d)∃x in D such that ∀y in E, x ≤ y. Negation: __ in D, __ in E such that x __ y.

∀x, ∃y, >

Write a negation for the statement. ∀ computer program P, if P compiles without error messages, then P is correct.

∃ a computer program P such that P compiles without error messages but P is not correct.

Consider the following statement. Everybody trusts somebody. (b) Write a negation for the statement.

∃ a person x such that ∀ person y, x does not trust y

Let D be the set of all quadrilaterals, and consider the predicates: R(x) : x is rectangle S(x) : x is a square Rewrite the following in symbolic form (that is, with quantifiers, variables, and the predicates above): Some rectangles are not squares

∃ x ∈ D such that R(x) ∧ ~S(x)

Let D be the set of all quadrilaterals, and consider the predicates: R(x) : x is rectangle S(x) : x is a square Rewrite the following in symbolic form (that is, with quantifiers, variables, and the predicates above): There exists quadrilateral that is a square.

∃x ∈ D such that S(x)

For each of the following, write the negation and determine if the statement or its negation is true: Statement: ∀x ∈ ℝ, ∃y ∈ R such that (x+y)^1/2 = x^(1/2) + y(1/2) Which is true

∃x ∈ ℝ such that ∀y ∈ ℝ, (x+y)^1/2 ≠ x^(1/2) + y^(1/2), statment

Assume x is a particular real number and use De Morgan's laws to write the negation for the statement. −4 < x < 3

−4 ≥ x or x ≥ 3

What is an argumnet?

A sequence of statements

"If p then q" What is q?

conclusion

Write a formal negation for each of the following statements. (b) ∀ computer c, c has a CPU.

∃ a computer c such that c does not have a CPU.

Negate " If x=4, the x^2=12"

"x=4 and x^2≠12"

Let p, q, and r be statements. Fully distribute the following negations: ~((~p ∨ q) ∧r)

(p ∧ ~q) ∨ ~r

~ ((p ∧ ~q) ∨ r) ≡

(p→q) ∧ r

(x ≤ b) is logically equivallent to?

(x < b) ∨ (x = b)

x ≤ b means?

(x < b) ∨ (x = b)

Use the unique factorization theorem to write the following integers in standard factored form. (c) 2,205

3^2 * 5^1 * 7^2

Show that the following number is rational by writing it as a ratio of two integers. 5.8053

58053/10000

The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You are visiting the island and have the following encounters with natives. (a) Two natives A and B address you as follows. A says: Both of us are knights. B says: A is a knave. What are A and B?

A is a knave and B is a knight.

The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You are visiting the island and have the following encounters with natives. (b) Another two natives C and D approach you but only C speaks. C says: Both of us are knaves.

C is a knave and D is a knight.

Modus Ponens

If P then Q P Therefore Q

Consider the following statement. Sam will be allowed on Signe's racing boat only if he is an expert sailor. Rewrite the statement in if-then form in two ways, one of which is the contrapositive of the other. Use the formal definition of "only if." (Select all that apply.)

If Sam is not an expert sailor, then he will not be allowed on Signe's racing boat. If Sam is allowed on Signe's racing boat, then he is an expert sailor.

Rewrite each statement without using quantifiers or variables. Indicate which are true and which are false. Let the domain of x be the set of geometric figures in the plane, and let Square(x) be "x is a square" and Rect(x) be "x is a rectangle." (a) ∃x such that Rect(x) ∧ Square(x)

There are some geometric figures that are both rectangles and squares

Rewrite each statement without using quantifiers or variables. Indicate which are true and which are false. Let the domain of x be the set of geometric figures in the plane, and let Square(x) be "x is a square" and Rect(x) be "x is a rectangle." (b) ∃x such that Rect(x) ∧ ~Square(x)

There are some geometric figures that are rectangles, but are not squares

Rewrite each statement without using variables or the symbol ∀ or ∃. Indicate whether the statement is true or false. (b) ∃ a real number r such that ∀ nonzero real numbers s, rs = 1.

There is a real number whose product with every nonzero real number equals 1.

Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous. (b) All graphs are connected.

There is at least one graph that is disconnected

Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous. (a) All dogs are friendly.

There is at least one unfriendly dog.

(b) ∃x such that Rect(x) ∧ ~Square(x) This statement is (T or F)

True

(c) ∀x, Square(x) → Rect(x) The statement is (T or F)

True

Consider the argument: If the sum of two real numbers is 0, then one is negative the other. x and y are two particular real numbers that sum to 0. Therefore, x = -y Choose the one that accurately describes the argument form: a. Valid, by universal modus ponens. b. Valid, by universal modus tollens. c. Invalid, converse error. d. Invalid, inverse error.

a

Consider the argument: If an integer is prime, then it is eihter 2 or odd. The number n is a particular numbr that is not prime. Therefore, n is neither 2 nor odd. Choose the one that accurately describes the argument form: a. Valid, by universal modus ponens. b. Valid, by universal modus tolllens. c. Invalid, converse error. d. Invald, inverse error.

d

odd integer n

n = 2k +1 for some integer k

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (d)∃x in D such that ∀y in E, x ≤ y. Negation: ∀x in D, ∃y in E such that x > y. Which is true: ( the given statement or the negation)

the given statement

Let m = "Haley is a math major" d= "Haley is taking Discrete Mathematics" c = "Haley is taking Calculus I" Rewrite the following statements in symbolic form: (d) "If Haley is not taking both Calculus I and Discrete Math, then she is not a Math Major."

~m ∧ (~d ∧ ~c) ≡ m ∧ ~(d ∨ c)

De Morgan's Law ~ (p ∨ q) ≡

~p ∧ ~q

What is the inverse of p→q

~p→~q

For each of the following, write the negation and determine if the statement or its negation is true: Statement: ∀a ∈ ℝ, ∃b ∈ ℝ such that ab = a Which is true

∃a ∈ ℝ such that ∀b ∈ ℝ, ab ≠ a, staement

Write a formal negation for each of the following statements: ∀n ∈ ℕ, if n is even, then n is not prime

∃n ∈ ℕ such that if n is even and n is prime

Write the statement in symbolic form using the symbols ~, ∨, and ∧ and the indicated letters to represent component statements. Let p = "x > 6," q = "x = 6," and r = "11 > x. 11 > x > 6

r ∧ p

Assume x is a particular real number and use De Morgan's laws to write the negation for the statement. x ≤ −4 or x > 4

x > −4 and x ≤ 4

Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) If x is nonnegative, then x is positive or x is 0. (a) Negation

x is nonnegative and x is not positive and x is not 0.

Determine whether the statements in A and B are logically equivalent. Statement A: Bob is both a math and computer science major and Ann is a math major, but Ann is not both a math and computer science major. Statement B: It is not the case that both Bob and Ann are both math and computer science majors, but it is the case that Ann is a math major and Bob is both a math and computer science major. Let b be "Bob is a double math and computer science major," m be "Ann is a math major," and a be "Ann is a double math and computer science major." Create a truth table for the two statements. Write the statements in symbolic form using the symbols ~, ∨, and ∧ and the indicated letters to represent component statements. Statement B

~ (b ∧ a) ∧ (m ∧ b)

Write each of the three statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explanation to show that you understand what it means for statements to be logically equivalent. Let p represent "It walks like a duck," q represent "It talks like a duck," and r represent "It is a duck." (c) If it does not walk like a duck and it does not talk like a duck, then it is not a duck.

~p ∧ ~q → ~r

What is the contrapositive of p→q

~q→~p

Write a formal negation for each of the following statements. (d) ∃ a band b such that b has won at least 10 Grammy awards.

∀ band b, b has won fewer than 10 Grammy awards.

Rewrite the following statement in the form "∀ _____ x, if _____ then _____." All Java programs have at least 5 lines.

∀ computer programs x, if x is written in Java then x has at least 5 lines

Rewrite each of the following statements in the form "∀ _____ x, _____." (a) All dinosaurs are extinct

∀ dinosaurs x, x is extinct

Rewrite each of the following statements in the form "∀ _____ x, _____." (e) The number 2,147,581,953 is not equal to the square of any integer.

∀ integers x, x² does not equal 2,147,581,953

Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (a) There is an engineering student who is a math major.

∃ s { D such that E(s) ∧ M(s)

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (b) ∃x in D such that ∀y in E, x + y = −y. Negation: __ in D, in__ E such that x + y __ −y.

∀x, ∃y, ≠

Consider the following statement. Somebody trusts everybody. (a) Rewrite the statement formally using quantifiers and variables.

∃ a person x such that ∀ person y, x trusts y.

Write a formal negation for each of the following statements. (a) ∀ string s, s has at least one character.

∃ a string s such that s does not have any characters.

Write a negation for the statement. ∀ integer d, if 8/d is an integer then d = 4.

∃ an integer d such that 8/d is an integer and d ≠ 4.

Write a negation for the statement. ∀ integer n, if n is divisible by 6, then n is divisible by 2 and n is divisible by 3.

∃ an integer n such that n is divisible by 6 and either n is not divisible by 2 or n is not divisible by 3

Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (d) Some computer science students are also math majors

∃ s { D such that C(s) ∧ M(s)

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (c) ∀x in D, ∃y in E such that xy ≥ y. Negation: __ in D such that __ in E, xy __ y.

∃x, ∀y, <

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (a) ∀x in D, ∃y in E such that x + y = 1. Negation: __ in D such that __ in E, x + y __ 1.

∃x, ∀y, ≠

Write each of the three statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explanation to show that you understand what it means for statements to be logically equivalent. Let p represent "It walks like a duck," q represent "It talks like a duck," and r represent "It is a duck." (b) Either it does not walk like a duck or it does not talk like a duck, or it is a duck.

(~p ∨ ~q) ∨ r

Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (e) Some computer science students are engineering students and some are not

(∃ s { D such that C(s) ∧ E(s)) ∧ (∃ s { D such that C(s) ∧ ~E(s))

Find a counterexample to show that the statement is false. ∀ a ℤ, (a − 1)a is not an integer. a =

-1

The following statement is true: "∀ nonzero number x, ∃ a real number y such that xy = 1." For each x given below, find a y to make the predicate "xy = 1" true. (b) x = −1 Let y =

-1

Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous. (c) Some suspicions were substantiated.

All suspicions were unsubstantiated.

Let D = E = {−2, −1, 0, 1, 2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. (b) ∃x in D such that ∀y in E, x + y = −y. Negation: ∀x in D, ∃y in E such that x + y ≠ −y. Which is true: ( the given statement or the negation)

The negation

Use De Morgan's laws to write the negation for the statement. The train is late or my watch is fast.

The train is not late and my watch is not fast.

The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You are visiting the island and have the following encounters with natives. (d) Finally, you meet a group of six natives, U, V, W, X, Y, and Z, who speak to you as follows. U says: None of us is a knight. V says: At least three of us are knights. W says: At most three of us are knights. X says: Exactly five of us are knights. Y says: Exactly two of us are knights. Z says: Exactly one of us is a knight. Which are knaves? (Select all that apply.)

U, V, X, Z

The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You are visiting the island and have the following encounters with natives. (d) Finally, you meet a group of six natives, U, V, W, X, Y, and Z, who speak to you as follows. U says: None of us is a knight. V says: At least three of us are knights. W says: At most three of us are knights. X says: Exactly five of us are knights. Y says: Exactly two of us are knights. Z says: Exactly one of us is a knight. Which are knights? (Select all that apply.)

W, Y

The argument below may be valid or exhibit the converse or the inverse error. Use symbols to write the logical form of the argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. Sandra knows Java and Sandra knows C++. ∴ Sandra knows C++. Let p = "Sandra knows Java" and q = "Sandra knows C++."

form: p ∧ q valid, specialization ∴ q