Chapter 3 - The Logic of Quantified Statements

Write an informal negation for each of the following statement. All dogs are friendly.

There is at least one dog that is unfriendly.

A negation for "For all x, if x has property P then x has property Q" is "______."

This is an x such that x has property P and x does not have property Q.

Consider the statement "∃x such that ∀y, P(x, y), a property involving x and y, is true." A negation for this statement is:

∀x such that ∃y, P(x,y) is false.

What is the converse of the following statement: ∀x ∈ D, if P(x) then Q(x).

∀x ∈ D, if Q(x) then P(x).

What is the inverse of the following statement: ∀x ∈ D, if P(x) then Q(x).

∀x ∈ D, if ∼P(x) then ∼Q(x).

What is the contrapositive of the following statement: ∀x ∈ D, if P(x) then Q(x).

∀x ∈ D, if ∼Q(x) then ∼P(x).

Convert "For all x in the set of all human beings, x is mortal" using universal quantifiers.

∀x ∈ H, x is mortal

What is the negation of: ∃x(Square(x) ∧ Black(x)).

∀x(∼Square(x) ∨ ∼Black(x)) This uses the De'Morgan laws

Rewrite the following statement in the two forms, all squares are rectangles.: "∀x, if _____ then ______" and "∀ _______ x, ______":

∀x, if x is a square then it is a rectangle. ∀ squares x, x is a rectangle.

Rewrite the following statement as quantified conditional statements. Do not use the word necessary or sufficient. Squareness is a sufficient condition for rectangularity.

∀x, if x is a square, then x is a rectangle.

Write a formal negation for each of the following statement: ∀ fish x, x has gills.

∃ a fish x, such that x does not have gills.

Rewrite the following statements in the two forms "∃ _____ x such that _____ " and "∃x such that _____ and _____ ." Some hatters are mad.

∃ a hatter x such that x is mad ∃x such that x is a hatter and x is mad

Write a formal negation of the following sentence: ∀ people p, if p is blond then p has blue eyes.

∃ a person p such that p is blond and p does not have blue eyes.

Write a negation of the following sentence: ∀ primes p, p is odd

∃ a prime p such that p is not odd.

Rewrite the statement formally. Use quantifiers and variables. Some programs are structured.

∃ a program p such that p is structured.

Translate the informal statement to formal: There is a real number with no reciprocal.

∃ a real number c such that ∀ real numbers d,cd ̸= 1.

Rewrite each of the following in the form: "∃ _______ x such that ______." Some exercises have answers.

∃ an exercise x such that x has an answer.

Write a negation for the following statement: ∀ integers a, b and c, if a−b is even and b−c is even, then a − c is even.

∃ integers a, b and c if a-b is even and b-c is even then a-c is not even.

What is the negation of: ∼(∀ x in D, ∃y in E such that P(x, y))

∃x in D such that ∀y in E, ∼P(x, y).

What is the negation of ∀x in D, Q(x)?

∃x in D such that ∼Q(x).

What is the negation of a universal conditional statement? ∼(∀x, if P(x) then Q(x))

∃x such that P(x) and ∼Q(x)

What is the negation of: ∀x(Circle(x) → ∃y(Square(y) ∧ SameColor(x, y))).

∃x(Circle(x) ∧ ∀y(∼Square(y) ∨ ∼SameColor(x, y)))

What is the negation of: ∀x(Circle(x) → Above(x, f ))

∃x(Circle(x) ∧ ∼Above(x, f )) This uses the law of negating and if then statement.

Consider the statement "∀x,∃y such that P(x,y), a property involving x and y, is true." A negation for this statement is:

∃x, such that ∀y the property P(x,y) is false.

Write a negation for the following statement: ∀x∈R, if x(x+1)>0 then x>0 or x<−1.

∃x∈R, if x(x+1) > 0 then x <= 0 and x >= -1

Find counter example of the following statement: ∀x∈R, x > 1/x.

Simply 1 in this scenario would work since 1 is not > 1.

Write a negation for each of the following statement: Any valid argument has a true conclusion.

There exists one valid argument that does not have a true conclusion.

Use the facts that the negation of a ∀ statement is a ∃ statement and that the negation of an if-then statement is an and statement to rewrite each of the statements without using the word necessary or sufficient. Being divisible by 8 is not a necessary condition for being divisible by 4.

There is a number that is divisible by 4 and is not divisible by 8.

Use the facts that the negation of a ∀ statement is a ∃ statement and that the negation of an if-then statement is an and statement to rewrite each of the statements without using the word necessary or sufficient. Having a large income is not a sufficient condition for a person to be happy.

There is a person who has a large income and is not happy.

Rewrite the following statement using an equivalent non formal definition: ∃m ∈ Z+such that m^2 = m

There is a positive integer whose square is equal to itself.

"∀x,r(x) only if s(x)" means

"∀x, if ∼s(x) then ∼r(x)" or, equivalently, "∀x, if r(x) then s(x)."

What is the negation of: ∼(∃x in D such that ∀y in E, P(x, y))

∀x in D,∃y in E such that ∼P(x, y).

A negation for "All R have property S" is "There is _____ R that _____."

"There is some R that does not have property S."

"∀x,r(x) is a sufficient condition for s(x)" means

"∀x, if r(x) then s(x)."

"∀x,r(x) is a necessary condition for s(x)" means

"∀x, if ∼r(x) then ∼s(x)" or, equivalently, "∀x, if s(x) then r(x)."

Find the truth set of each predicate: 6/d is an integer, domain: Z

-6, -3, -2, -1, 1, 2, 3, 6

Let B(x) be "−10 < x < 10." Find the truth set of B(x) for the domain Z

-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Write a negation for the following statement: If the square of an integer is odd, then the integer is odd.

At least one integer has an odd square but is not itself odd.

Write a negation for the following statement: Sets A and B do not have any points in common.

Sets A and B have at least one point in common.

Let D={−48,−14,−8,0,1,3,16,23,26,32,36}. Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. ∀x ∈ D, if x is even then x ≤ 0.

False, 16 and 26 are counter examples.

Rewrite the following statement using an equivalent non formal definition: ∀x∈R,x^2 ≥0

All real numbers have nonnegative squares.

Rewrite the following statement using an equivalent non formal definition: ∀x∈R,x^2 does not equal -1

All real numbers have squares that do not equal -1.

Write an informal negation for each of the following statement. Some suspicions were substantiated.

All suspicions were unsubstantiated.

What is a predicate?

A sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.

A statement of the form ∀x ∈ D, Q(x) is true if, and only if,Q(x)is __________ for ______ .

A statement of the form ∀x ∈ D, Q(x) is true if, and only if,Q(x) is true for every x in D.

A statement of the form ∃x ∈ D such that Q(x) is true if, and only if, Q(x) is ______ for _____.

A statement of the form ∃x ∈ D such that Q(x) is true if, and only if, Q(x) is true for at least one x in D.

Write informal negation of the following sentence: Some computer hackers are over 40.

All computer hackers are under 40.

Determine whether the proposed negation is correct. If it is not, write a correct negation. Statement: For all integers n, if n^2 is even then n is even. Proposed negation: For all integers n, if n^2 is even then n is not even.

False, the negation would be: There is an integer n, if n^2 is even then n is not even.

Write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. If the square of an integer is odd, then the integer is odd.

Converse: Inverse: Contrapositive:

Write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. ∀ integers a, b and c, if a−b is even and b−c is even, then a − c is even.

Converse: Inverse: Contrapositive:

Write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. ∀ real numbers x, if x^2 ≥ 1 then x >0.

Converse: Inverse: Contrapositive:

Write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. ∀x∈R, if x(x+1) > 0 then x>0 or x<−1.

Converse: Inverse: Contrapositive:

Rewrite the following statement informally in at least two different ways without using variables or the symbol ∀ or the words "for all." ∀ real numbers x , if x is positive, then the square root of x is positive.

For all real numbers that are positive the the square root is positive. The square root of a positive real number is positive.

Rewrite the following statement informally, without quantifiers or variables. ∀x∈R, if x>2 then x^2 >4.

For all real numbers x, if x is greater than 2 its square is greater than 4.

Rewrite each of the following statements in the two forms "∀x, if____ then ____ " All equilateral triangles are isosceles

For all x, if an equilateral triangle then x isosceles.

The contrapositive of "For all x, if x has property P then x has property Q" is "______."

For all x, if x does not have property Q then x does not have property P.

The inverse of "For all x, if x has property P then x has property Q" is "______ ."

For all x, if x does not have property p then x does not have property q.

The converse of "For all x, if x has property P then x has property Q" is "______."

For all x, if x has property Q then x has property P.

Rewrite the following statements in the form: "∀ ______ x, if _____ then _____." All java programs have at least 5 lines.

For all x, if x is java then then x has a least 5 lines.

Rewrite the following statement formally. Then write formal and informal negations. No politicians are honest.

Formal: ∀ politicians x, x is not honest Formal Negation: ∃ a politician x, such that x is honest Informal Negation: It exists a politician that is honest.

If P(x) is a predicate with domain D, the truth set of P(x) is denoted _____. We read these symbols out loud as ________.

If P(x) is a predicate with domain D, the truth set of P(x) is denoted {x ∈ D | P(x)}. We read these symbols out loud as the set of all x in D such that P(x).

Rewrite each statement in the if-them form: Earning a grade of C− in this course is a sufficient condition for it to count toward graduation.

If a person earns a C- in this course the the course counts toward graduation.

Rewrite the statements in each pair in if-then form and indicate the logical relationship between them. All the children in Tom's family are female. All the females in Tom's family are children.

If a person is a child in Tom's family, then the person is a female. If a person is a female in Tom's family then the person is a child. The second statement is a converse of the first.

Rewrite each statement in the if-them form: Being on time each day is a necessary condition for keeping this job.

If a person is not on time each day then they won't keep their job.

The following statements is true. Write the converse of the statement, and give a counterexample showing that the converse is false. If n is any prime number that is greater than 2, then n + 1 is even.

If n+1 is even then n is a prime number that is greater than 2.

A negation for "Some R have property S" is:

No R have property S.

Write informal negation of the following sentence: All computer programs are finite.

It exists a computer program that is infinite.

Write a negation for the following sentence: ∀ real numbers x, if x >3 then x^2 >9.

It exists a real number x, if x > 3 then x^2 less than or equal to 9.

Write a negation for the following statement: ∀ real numbers x, if x^2 ≥1 then x >0.

It exists a real number x, that if x^2 ≥ 1 then x < 0.

Write a formal negation of the following sentence: If a computer program has more than 100,000 lines, then it contains a bug.

There is at least one computer program that has more than 100,000 lines and does not contain a bug.

Some ways to express the symbol ∀ in words are ______.

Some ways to express the symbol ∀ in words are for all, for any.

Some ways to express the symbol ∃ in words are ______.

Some ways to express the symbol ∃ in words are it exists, at least one.

Let Q(n) be the predicate "n is a factor of 8." Find the truth set of Q(n) if the domain of n is the set Z+ of all positive integers

The truth set is {1,2,4,8} because these are exactly the positive integers that divide 8 evenly.

Consider the following sequence of digits: 0204. A person claims that all the 1's in the sequence are to the left of all the 0's in the sequence. Is this true? Justify your answer. (Hint: Write the claim formally and write a formal negation for it. Is the negation true or false?)

The negation would be It exists a number 1 that is not to the left of the 0's in the sequence. The negation is false because the sequence does not contain a 1.

Write informal negation of the following sentence: The number 1,357 is divisible by some integer between 1 and 37.

The number 1,357 is not divisible by any integer between 1 and 37.

Determine whether the proposed negation is correct. If it is not, write a correct negation. Statement: The sum of any two irrational numbers is irrational. Proposed negation: The sum of any two irrational numbers is rational.

The proposed negation is not correct. The correct negation should be: There are at least two irrational numbers whose sum is rational.

Let Q(n) be the predicate "n is a factor of 8." Find the truth set of Q(n) if the domain of n is the set Z of all integers.

The truth set is {1,2,4,8,−1,−2,−4,−8} because the negative integers −1,−2,−4, and −8 also divide into 8 without leaving a remainder.

To establish the truth of a statement of the form "∃x in D such that ∀y in E, P(x, y)," you need to find ______ so that no matter what ______ a person might subsequently give you, _____ will be true.

To establish the truth of a statement of the form "∃x in D such that ∀y in E, P(x, y)," you need to find an element of x of D so that no matter what y in E a person might subsequently give you, P (x,y) will be true.

Let D={−48,−14,−8,0,1,3,16,23,26,32,36}. Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. ∀x ∈ D, if x is odd then x >0.

True

Find counter example of the following statement: ∀ positive integers m and n,m·n ≥ m + n.

When m=1 and n=1 then m·n = 1 and m+n equals 2 which contradicts the formula.

Rewrite the following statements informally in at least two different ways without using variables or quantifiers. a. ∀ rectangles x , x is a quadrilateral. b. ∃ a set A such that A has 16 subsets.

a. Every rectangle is a quadrilateral. b. At least one set has 16 subsets.

Which of the following is a negation for" All discrete mathematics students are athletic"? More than one answer may be correct. a. There is a discrete mathematics student who is nonathletic. b. All discrete mathematics students are nonathletic. c. There is an athletic person who is a discrete mathematics student. d. No discrete mathematics students are athletic. e. Some discrete mathematics students are nonathletic. f. No athletic people are discrete mathematics students.

a, e

Consider the following statement: ∀ basketball players x , x is tall. Which of the following are equivalent ways of expressing this statement? a. Every basketball player is tall. b. Among all the basketball players, some are tall. c. Some of all the tall people are basketball players. d. Anyone who is tall is a basketball player. e. All people who are basketball players are tall. f. Anyone who is a basketball player is a tall person.

a, e, f

Let Q(n) be the predicate "n^2 ≤ 30." Find the truth set of Q(n) if the domain of n is Z, the set of all integers.

a. -5, -4, -3, -2, -1 ,0, 1, 2, 3, 4, 5

Let S be the set of students at your school, let M be the set of movies that have ever been released, and let V (s, m) be "student s has seen movie m." Rewrite each of the following statements without using the symbol ∀, the symbol ∃, or variables. a. ∃s ∈ S such that V (s, Casablanca). c. ∀s ∈ S,∃m ∈ M such that V(s,m). d. ∃m ∈ M such that ∀s ∈ S, V(s,m).

a. At least one student has watched Casablanca. c. All students have watched at least one movie. d. There is one movie that all students have watched.

Indicate which of the following statements are true and which are false. Justify your answers as best as you can. a. Every integer is a real number. b. 0 is a positive real number. c. For all real numbers r, −r is a negative real number. d. Every real number is an integer.

a. True, the statement is true. The integers correspond to certain points on the number line and the real numbers correspond to all the points on the number line. b. False, 0 is neither positive or negative. c. False, since - (-2) equals a positive 2. d. False, for instance 1/2 is a real number but not an integer.

Rewrite each of the following statements in the form "∀ _________ x, _______ ." a. All dinosaurs are extinct. c. No irrational numbers are integers. e. The number 2,147,581,953 is not equal to the square of any integer.

a. ∀ dinosaurs x, x is extinct. c. ∀ irrational number x, it is not an integer. e. ∀ integers x, x^2 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. b. Every computer science student is an engineering student. e. Some computer science students are engineering students and some are not.

a. ∃s ∈ D such that E(s) and M(s) b. ∀s ∈ D if C(s) then E(s) e. ∃s ∈ D such that C(s) ^ E(s) v C(s) ^ ~ E(s)

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

b, d, e

Rewrite the statement formally. Use quantifiers and variables. No dogs have wings.

∀ dogs d, d does not have wings.

Convert "All human beings are mortal" using universal quantifiers.

∀ human beings x , x is mortal.

Write a formal negation for each of the following statement: ∃ a movie m such that m is over 6 hours long.

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

Translate the informal statement to formal: Every nonzero real number has a reciprocal.

∀ nonzero real numbers u, ∃ a real number v such that uv = 1.

Rewrite the following statement as quantified conditional statements. Do not use the word necessary or sufficient. Being at least 35 years old is a necessary condition for being President of the United States.

∀ people x , if x is younger than 35, then x cannot be President of the United States.

Rewrite each of the following statements in the form ∀ ________, if ______ then _____ . If a real number is an integer, then it is a rational number.

∀ real numbers x, if x is an integer then x is a rational number.

Write a negation of the following sentence: ∃ a triangle T such that the sum of the angles of T equals 200 degrees.

∀ triangles T, the sum of the angles T does not equal 200 degrees.

Rewrite the statement formally. Use quantifiers and variables. All triangles have three sides.

∀ triangles t , t has three sides.

Rewrite each of the following statements in the form ∀ ________, if ______ then _____ . All bytes have eight bits.

∀ x, if x is a byte then x has 8 bits.

Rewrite each of the following statements in the form ∀ ________, if ______ then _____ . No fire trucks are green.

∀ x, if x is a firetruck then x is not green.

What is the negation of ∃x in D such that Q(x)

∀x in D, ∼Q(x)