Unit 1 progress check and review
Consider the following propositions. Let S(x) denote "x has a good attitude," where the domain is the set of people in the class. Express the negation of the proposition "There is someone in this class who does not have a good attitude" using quantifiers, and then express the negation in English.
∀xS(x); "Everyone in this class has a good attitude."
Consider the following propositions. Let S(x) denote "x obeys the speed limit," where the domain is the set of all drivers. Express the negation of the proposition "Some drivers do not obey the speed limit" using quantifiers, and then express the negation in English.
∀xS(x); All drivers obey the speed limit.
Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. The absolute value of the product of two integers is the product of their absolute values. (Check all that apply.)
∀x∀y(∣∣xy∣∣=∣∣x∣∣∣∣y∣∣) The statement "The absolute value of the product of two integers is the product of their absolute values." can be expressed as ∀x∀y(∣∣xy∣∣=∣∣x∣∣∣∣y∣∣)
Let the domain of discourse be the set of all students in your class. If L(x, y) means x has learned programming language y, express the statement "All students in this class have learned at least one programming language" using quantifiers.
∀x∃y L(x, y)
Let the domain of discourse be the set of all students in your class. If P(x, y) means x plays the sport y, express the statement "Every student in this class plays some sport" using quantifiers.
∀x∃yP(x, y)
Let the domain of discourse be the set of all students in your class. If C(x, y, z) means persons x and y have chatted with each other in chat group z, express the statement "Every student in this class has chatted with at least one other student in at least one chat group" using quantifiers.
∀x∃y∃z(x ≠ y ∧ C(x, y, z))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everyone can be fooled by somebody.
∀y∃xF(x, y)
Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬ ∀ x ∃ yP(x, y)
∃ x ∀ y ¬P(x, y)
There is someone who can fool exactly one person besides himself or herself.
∃ x ∃ y (x ≠ y ∧ F(x, y) ∧ ∀ z ((F(x, z) ∧ z ≠ x) → z = y))
Express the negations of each of these statements so that all negation symbols immediately precede predicates. ∀ y ∃ x ∃ z(T (x, y, z) ∨ Q(x, y))
∃ y ∀ x ∀ z(¬T (x, y, z) ∧ ¬Q(x, y))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. There is exactly one person whom everybody can fool.
∃ y( ∀ xF(x, y) ∧ ∀ z( ∀ xF(x, z) → z = y))
Let the domain of discourse be the set of all students in your class. If V(x, y) means x has visited state y, express the statement "Some student in this class has visited Alaska but has not visited Hawaii" using quantifiers.
∃x(V(x, Alaska) ∧¬V(x, Hawaii))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Nancy can fool exactly two people.
∃ y1 ∃ y2(F(Nancy, y1) ∧ F(Nancy, y2) ∧ y1 ≠ y2 ∧ ∀ y(F(Nancy, y) → (y = y1 ∨ y = y2)))
Let the statements "If I work, it is either sunny or partly sunny," "I worked last Monday or I worked last Friday," "It was not sunny on Tuesday," and "It was not partly sunny on Friday." be the given premises. The conclusion that can be drawn from the given premises is "It was neither sunny nor partly sunny on Friday."
FALSE
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃ yW(José Orez, y)
José Orez has visited at least one website.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. C(Randy Goldberg, CS 252)
Randy Goldberg is enrolled in CS 252.
Determine whether the given compound propositions is satisfiable. (p ∨ q ∨ ¬r) ∧ (p ∨ ¬q ∨∨ ¬s) ∧∧ (p ∨∨ ¬r ∨∨ ¬s) ∧∧ (¬p ∨∨ ¬q ∨∨ ¬s) ∧∧ (p ∨∨ q ∨∨ ¬s) The compound proposition (p ∨∨ q ∨∨ ¬r) ∧∧ (p ∨∨ ¬q ∨∨ ¬s) ∧∧ (p ∨∨ ¬r ∨∨ ¬s) ∧∧ (¬p ∨∨ ¬q∨∨ ¬s) ∧∧ (p ∨∨ q ∨∨ ¬s) is
SATISFIABLE
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. W(Sarah Smith, www.att.com)
Sarah Smith has visited www.att.com.
Identify the error or errors in this argument that supposedly shows that if ∀x(P(x)∨Q(x)) is true then ∀x P(x) ∨ ∀x Q(x)) is true. Identify the error in steps 3 and 5. StepReason1. ∀x(P(x) ∨∨ Q(x))Premise2. P(c) ∨∨ Q(c) for arbitrary cUniversal instantiation from (1)3. P(c) for arbitrary cSimplification from (2)4. ∀xP(x)Universal generalization from (3)5. Q(c) for arbitrary cSimplification from (2)6. ∀xQ(x)Universal generalization from (5)7.(∀xP(x)) ∨∨ (∀xQ(x))Disjunction from (4) and (6)
Simplification applies to conjunctions, not disjunctions.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃x(C(x, Math 222) ∧ C(x, CS 252))
Some student is enrolled simultaneously in Math 222 and CS 252.
Translate these system specifications into English, where the predicate S(x, y) is "x is in state y" and where the domain for x and y consists of all systems and all possible states, respectively. ∃xS(x, open) ∨ ∃xS(x, diagnostic)
Some system is open, or some system is in a diagnostic state.
Translate these system specifications into English, where the predicate S(x, y) is "x is in state y" and where the domain for x and y consists of all systems and all possible states, respectively. ∃xS(x, open)
Some system is open.
Translate these system specifications into English, where the predicate S(x, y) is "x is in state y" and where the domain for x and y consists of all systems and all possible states, respectively. ∃x¬S(x, available)
Some system is unavailable.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃xC(x, Math 695)
Someone is enrolled in Math 695.
Fill the provided truth tables and check whether the given conditional statements are a tautology or contradiction or contingency. Is the conditional statement ¬(p → q) → p a tautology?
Yes it is tautology
For each of these arguments, determine whether the argument is correct or incorrect along with the reason why the argument form is valid or invalid. All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic.
correct: use of universal instantiation and modus ponens
For each of these arguments, determine whether the argument is correct or incorrect along with the reason why the argument form is valid or invalid. Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day.
correct: use of universal instantiation and modus tollens
Let the argument be "Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket." c(x): x is in this class. r(x): x owns a red convertible t(x): x has gotten a speeding ticket. Identify the rule of inference that is used to arrive at the conclusion that c(Linda) ∧ t(Linda) from the hypothesis c(Linda) and t(Linda).
conjunction
Identify the compound proposition involving the propositional variables p, q, and r that is true when p and q are true and r is false, but is false otherwise.
p ∧ q ∧ ¬r
Find the dual of the given compound proposition. p ∨ ¬q
p ∧ ¬q
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Let the domain consist of all students in your class. Express the statement below in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Some student in your class has a cat and a ferret, but not a dog.
∃x(C(x) ∧ F(x) ∧ ¬D(x))
If we let the domain be all animals, and S(x) = "x is a spider", I(x) = " x is an insect", D(x) = "x is a dragonfly", L(x) = "x has six legs", E(x, y ) = "x eats y", then the premises be "All insects have six legs," (∀x (I(x)→ L(x))) "Dragonflies are insects," (∀x (D(x)→I(x))) "Spiders do not have six legs," (∀x (S(x)→¬L(x))) "Spiders eat dragonflies." (∀x, y (S(x) ∧ D(y)) → E(x, y))) The conditional statement "∀x, If x is an insect, then x has six legs" is derived from the statement "All insects have six legs" using _____.
universal instantiation
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. Identify the rule of inference that is used to arrive at the statement s(y) → w(y) from the statement ∀x(s(x) → w(x)).
universal instantiation
Let the argument be "Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year." r(x): x is one of the five roommates listed. d(x): x has taken a course in discrete mathematics. a(x): x can take a course in algorithms. Identify the rule of inference that is used to arrive at the conclusion that ∀x(r(x) → a(x)) from the hypothesis r(y) → a(y).
universal generalization
Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers. The difference of two negative integers is not necessarily negative.
¬ ∀ x ∀ y((x < 0) ∧ (y < 0) → (x − y < 0))
Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. The difference of two positive integers is not necessarily positive
¬ ∀ x ∀ y((x > 0) ∧ (y > 0) → (x − y > 0))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. There is no one who can fool everybody.
¬ ∃ x ∀ yF(x, y)
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. No one can fool both Fred and Jerry.
¬ ∃ x(F(x, Fred) ∧ F(x, Jerry))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. No one can fool himself or herself.
¬ ∃ xF(x, x)
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Let the domain consist of all students in your class. Express the statement below in terms of C(x), D(x), F(x), quantifiers, and logical connectives. No student in your class has a cat, a dog, and a ferret.
¬∃x(C(x) ∧ D(x) ∧ F(x))
Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers. The absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers
∀ x ∀ y ( |x+y| ≤ |x| + |y| )
Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. The sum of the squares of two integers is greater than or equal to the square of their sum.
∀ x ∀ y (x^2 + y62 ≥ (x + y)^2)
Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. The sum of two negative integers is negative.
∀ x ∀ y((x < 0) ∧ (y < 0) → (x + y < 0))
Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers. The product of two negative integers is positive.
∀ x ∀ y((x < 0) ∧ (y < 0) → (xy > 0))
Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers. The average of two positive integers is positive.
∀ x ∀ y((x > 0) ∧ (y > 0) → ((x + y)/2 > 0))
Express the negations of each of these statements so that all negation symbols immediately precede predicates. ∃ x ∃ y(Q(x, y) ↔ Q(y, x))
∀ x ∀ y(Q(x, y) ↔ ¬Q(y, x))
Express the negations of each of these statements so that all negation symbols immediately precede predicates. ∃ x ∃ yP(x, y) ∧ ∀ x ∀ yQ(x, y)
∀ x ∀ y¬P(x, y) ∨ ∃ x ∃ y¬Q(x, y)
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everybody can fool somebody.
∀ x ∃ yF(x, y)
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everybody can fool Fred.
∀ xF(x, Fred)
Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬ ∃ y ∃ xP(x, y)
∀ y ∀ x¬P(x, y)
Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬ ∃ y( ∃ xR(x, y) ∨ ∀ xS(x, y))
∀ y( ∀ x¬R(x, y) ∧ ∃ x¬S(x, y))
Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬ ∃ y( ∀ x ∃ zT (x, y, z) ∨ ∃ x ∀ zU(x, y, z))
∀ y( ∃ x ∀ z¬T (x, y, z) ∧ ∀ x ∃ z¬U(x, y, z))
Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬ ∃ y(Q(y) ∧ ∀ x¬R(x, y))
∀ y(¬Q(y) ∨ ∃ xR(x, y))
Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Evelyn can fool everybody.
∀ yF(Evelyn, y)
Express the negations of each of these statements so that all negation symbols immediately precede predicates. ∃ z ∀ y ∀ xT (x, y, z)
∀ z ∃ y ∃ x¬T (x, y, z)
Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∃x(¬P(x)) ∧ ∀x((x < 0) → P(x)) without using quantifiers, instead use only negations, disjunctions, or conjunctions. Click on the choice or choices that correspond to a correct solution.
(¬P(1) ∨ ¬P(3) ∨ ¬P(5)) ∧ (P(-1) ∧ P(-3) ∧ P(-5))
How many rows appear in a truth table for this compound propositions? (p∧r∧s)∨(q∧t)∨(r∧¬t)(p∧r∧s)∨(q∧t)∨(r∧¬t) The number of rows needed for the truth table of the compound proposition (p∧r∧s)∨(q∧t)∨(r∧¬t)(p∧r∧s)∨(q∧t)∨(r∧¬t) is _____.
32
How many rows appear in a truth table for this compound propositions? (q → ¬p) ∨ (¬p → ¬q) The number of rows needed for the truth table of the compound proposition (q → ¬p) ∨ (¬p → ¬q) is ____.
4
Fill the provided truth tables and check whether the given conditional statements are a tautology or contradiction or contingency. The conditional statement (p∧q)→(p→q)(p∧q)→(p→q) is _____.
A tautology
For each of these statements, find a domain for which the statement is true or false. Identify the domains for which the statement "Everyone is studying discrete mathematics" is false. (Check all that apply.)
All the students who applied for a discrete mathematics course All students in the same year as you All the students in the world
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. Mark all valid translations. There may be more than one. ∀x∀y(((x ≥ 0) ∧ (y < 0)) → (x - y > 0)) (Check all that apply.)
An English statement for the given quantifier is "A nonnegative number minus a negative number is positive."
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃∃ xW(x, www.imdb.org)
At least one student has visited www.imdb.org.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃yC(Carol Sitea, y)
Carol Sitea is enrolled in some class.
Translate in two ways the given statement into logical expressions using predicates, quantifiers, and logical connectives. Let C(x) be the propositional function "x is in your class" and P(x) be "x has a cellular phone." Click and drag an expression into each domain so that the expression, subject to the domain, has the meaning "Everyone in your class has a cellular phone." Domain A: The domain consists of the students in your class. Domain B: The domain consists of all people.
DOMAIN A
Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement ∀∀ x(R(x) Λ H(x)) into English.
Every animal is a rabbit and hops.
F(x) is "x is funny," and the domain consists of all people. ∀x(C(x) → F(x))
Every comedian is funny.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∀x∀yQ(x, y)
Every student in the class has sent a message to every student in the class.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∀y∃xQ(x, y)
Every student in your class has been sent a message from at least one student in your class.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∀x∃yQ(x, y)
Every student in your class has sent an e-mail message to at least one student in your class.
Translate these system specifications into English, where the predicate S(x, y) is "x is in state y" and where the domain for x and y consists of all systems and all possible states, respectively. ∀x(S(x, malfunctioning) ∨ S(x, diagnostic))
Every system is either malfunctioning or in a diagnostic state.
Let the premises be the statements "Every student has an Internet account," "Homer does not have an Internet account," and "Maggie has an Internet account." The conclusion about Maggie is "Maggie is a student."
FALSE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∀ n ∀ m ∃ p(p = (m + n)/2)
FALSE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∃ m(n + m = 4 ∧ n − m = 1)
FALSE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∃ m(n^2 + m^2 = 6)
FALSE
Determine the truth value of the statement ∀x∃y(xy = 1) if the domain for the variables consists of the nonzero integers.
FALSE
Find the truth values of these statements if the domain consists of all integers. ∃!x(x = x + 1)
FALSE
Find the truth values of these statements if the domain consists of all integers. ∃!x(x > 1)
FALSE
Find the truth values of these statements if the domain consists of all integers. ∃!x(x2 = 1)
FALSE
If we let the domain be all animals, and S(x) = "x is a spider", I(x) = " x is an insect", D(x) = "x is a dragonfly", L(x) = "x has six legs", E(x, y ) = "x eats y", then the premises be "All insects have six legs," (∀x (I(x)→ L(x))) "Dragonflies are insects," (∀x (D(x)→I(x))) "Spiders do not have six legs," (∀x (S(x)→¬L(x))) "Spiders eat dragonflies." (∀x, y (S(x) ∧ D(y)) → E(x, y))) No conclusions can be drawn from the conditional statement "∀x, If x is an insect, then x has six legs" and the statement "Spiders do not have six legs" using modus tollens.
FALSE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? Q(1,1)
FALSE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∀ x ∀ yQ(x, y)
FALSE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∀ y ∃ xQ(x, y)
FALSE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∀∀ yQ(1, y)
FALSE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∃ xQ(x, 2)
FALSE
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. The premises are ∀x(s(x) → w(x)) and ∀x(s(x) ∧ c(x)).
FALSE
Let the argument be "Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket." c(x): x is in this class. r(x): x owns a red convertible t(x): x has gotten a speeding ticket. The formal expression of the statement "Everyone who owns a red convertible has gotten at least one speeding ticket" is ∀x(t(x) → r(x)).
FALSE
Let the premises be the statements "All foods that are healthy to eat do not taste good," "Tofu is healthy to eat," "You only eat what tastes good," "You do not eat tofu," and Cheeseburgers are not healthy to eat." The conclusion from the statements "For all x, if x is healthy to eat, then x does not taste good," "You only eat what tastes good," and "Cheeseburgers are not healthy to eat." is "Eat only cheeseburgers."
FALSE
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. The given premises are ∃x(c(x) ∧ f(x)), ∃x(f(x) → l(x)).
FALSe
Determine whether the given conditional statements is true or false. If 1 + 1 = 2, then dogs can fly.
False
Translate the given statement into English. ∀x∀y(((x ≥ 0) ∧ (y ≥ 0)) → (xy ≥ 0)) where x and y are real numbers.
For every real number x and real number y, if x and y are both nonnegative, then their product is nonnegative.
Translate the given statement into English. ∀x∀y∃z(xy = z) where x and y are real numbers.
For every real number x and real number y, there exists a real number z such that xy = z.
Translate the given statement into English. ∀x∃y(x < y) where x and y are real numbers.
For every real number x there exists a real number y such that xis less than y.
Let the premises be the statements "Every student has an Internet account," "Homer does not have an Internet account," and "Maggie has an Internet account." Identify the conclusion about Homer.
Homer is not a student.
Translate these specifications into English where F(p) is "Printer p is out of service," B(p) is "Printer p is busy," L(j) is "Print job j is lost," and Q(j) is "Print job j is queued." (∀pB(p) ∧ ∀jQ(j)) → ∃jL(j)
If every printer is busy and every job is queued, then some job is lost.
Translate these specifications into English where F(p) is "Printer p is out of service," B(p) is "Printer p is busy," L(j) is "Print job j is lost," and Q(j) is "Print job j is queued." ∀pB(p) → ∃jQ(j)
If every printer is busy, then there is a job in the queue.
Translate these specifications into English where F(p) is "Printer p is out of service," B(p) is "Printer p is busy," L(j) is "Print job j is lost," and Q(j) is "Print job j is queued." ∃j(Q(j) ∧ L(j)) → ∃pF(p)
If there is a job that is both queued and lost, then some printer is out of service.
Translate these specifications into English where F(p) is "Printer p is out of service," B(p) is "Printer p is busy," L(j) is "Print job j is lost," and Q(j) is "Print job j is queued." ∃p(F(p) ∧ B(p)) → ∃jL(j)
If there is a printer that is both out of service and busy, then some job has been lost.
Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. Experience with C++ or Java is required.
Inclusive or
If George does not have eight legs, then he is not a spider. George is a spider. Therefore George has eight legs. Find the argument form for the given argument and determine whether it is valid. (You must provide an answer before moving to the next part.) The argument form for the given argument is (Click to select) resolution hypothetical syllogism modus tollens modus ponens , and this argument is (Click to select) valid not valid .
MODUS TOLLENS VALID
Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∃xP(x) without using quantifiers, instead use only negations and disjunctions. Click on the choice or choices that correspond to a correct solution.
P(-5) ∨ P(-3) ∨ P(-1) ∨ P(1) ∨ P(3) ∨ P(5)
Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∀xP(x) without using quantifiers, instead use only negations, disjunctions, or conjunctions. Click on the choice or choices that correspond to a correct solution.
P(1) ∧ P(3) ∧ P(5) ∧ P(-5) ∧ P(-3) ∧ P(-1)
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∀ n ∃ m(n + m = 0)
TRUE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∀ n ∃ m(n2 < m)
TRUE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∀ m(n < m^2)
TRUE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∀ m(nm = m)
TRUE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∃ m(n + m = 4 ∧ n − m = 2)
TRUE
Determine the truth value of each of these statements if the domain for all variables consists of all integers. ∃ n ∃ m(n^2 + m^2 = 5)
TRUE
Determine the truth value of the statement ∀x∃y(xy = 1) if the domain for the variables consists of the nonzero real numbers.
TRUE
Determine the truth value of the statement ∀x∃y(xy = 1) if the domain for the variables consists of the positive real numbers.
TRUE
Find the dual of the given compound proposition. The dual of the compound position (p∧¬q)∨(q∧F)(p∧¬q)∨(q∧F) is (p∨¬q)∧(q∨T)(p∨¬q)∧(q∨T) .
TRUE
Find the truth values of these statements if the domain consists of all integers. ∃!x(x + 3 = 2x)
TRUE
For each of these statements, find a domain for which the statement is true or false. The statement "Everyone is older than 21 years" is (Click to select) false true in the domain "All United States senators."
TRUE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? Q(2,0)
TRUE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∀ x ∃ yQ(x, y)
TRUE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∃ x∃∃ yQ(x,y)
TRUE
Let Q(x,y) be the statement "x + y = x - y." If the domain for both variables consists of all integers, what are these truth values? ∃ y ∀ xQ(x, y)
TRUE
Let the argument be "Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year." r(x): x is one of the five roommates listed. d(x): x has taken a course in discrete mathematics. a(x): x can take a course in algorithms. The given premises are ∀x(r(x) → d(x)) and ∀x(d(x) → a(x)).
TRUE
Let the premises be the statements "I am either dreaming or hallucinating," "I am not dreaming," and "If I am hallucinating, I see elephants running down the road." The conclusion that follows from the statement "I am either dreaming or hallucinating," and "I am not dreaming" is "I am hallucinating."
TRUE
Let the statements "If I play hockey, then I am sore the next day," "I use the whirlpool if I am sore," and "I did not use the whirlpool" be the given premises. Modus tollens is used to arrive at the conclusion that I did not play hockey.
TRUE
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. Mark all valid translations. There may be more than one. ∀x∀y(((x ≠ 0) ∧ (y ≠ 0)) ↔ (xy ≠ 0)) (Check all that apply.)
The English statement for the given quantifier are: The product of two numbers is nonzero if and only if both factors are nonzero. A necessary and sufficient condition for a product of two numbers being nonzero is both factors being nonzero.
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. Mark all valid translations. There may be more than one. ∃x∀y(x + y = y) (Check all that apply.)
The English statements for the given quantifier are as follows: There exists an additive identity for the real numbers - a number that when added to every number does not change its value. There is a real number that has the property that adding it to a real number does not change the value of that number.
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. Mark all valid translations. There may be more than one. ∃x∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x - y > 0)) (Check all that apply.)
The English statements for the given quantifier are: The difference of two nonpositive numbers is not necessarily nonpositive. There are nonpositive numbers so that the first minus the second is positive.
Identify the mistake, if there is one, in the following argument. Let S(x, y) be "x is shorter than y." Given the premise ∃sS(s, Max), it follows that S(Max, Max). Then, by existential generalization it follows that ∃xS(x, x), so that someone is shorter than himself.
The argument is incorrect. We know that some s exists that makes S(s, Max) true, but we cannot conclude that Max is one such s.
Identify the mistake, if there is one, in the following argument. Let H(x) be "x is happy." Given the premise ∃xH(x), we conclude that H(Lola). Therefore, Lola is happy.
The argument is incorrect. We know that some x exists that makes H(x) true, but we cannot conclude that Lola is one such x.
Let the statements "If I play hockey, then I am sore the next day," "I use the whirlpool if I am sore," and "I did not use the whirlpool" be the given premises. Identify the conclusions that can be drawn from the given premises. (Check all that apply.)
The conclusions are as follows: I am not sore. I did not play hockey.
Find a common domain for the variables x, y, and z for which the statement ∀x∀y((x ≠ y) → ∀z((z = x) ∨ (z = y))) is true and another domain for which it is false.
The domain in which the given statement is true is any set having at most two elements. The domain in which the given statement is false is any set having more than two elements.
Identify the error or errors in this argument that supposedly shows that if ∃xP(x) ∧ ∃xQ(x) is true, then ∃x(P(x) ∧ Q(x)) is true. 1. ∃xP(x) ∧ ∃xQ(x) Premise2. ∃xP(x) Simplification from (1)3. P(c) Existential instantiation from (2)4. ∃xQ(x) Simplification from (1)5. Q(c) Existential instantiation from (4)6. P(c) ∧ Q(c) Conjunction from (3) and (5)7. ∃x(P(x) ∧ Q(x)) Existential generalization
The error is in step 5; we cannot assume that the c that makes P true and the c that makes Q true are the same.
Identify the error or errors in this argument that supposedly shows that if ∀x(P(x)∨Q(x)) is true then ∀x P(x) ∨ ∀x Q(x)) is true. StepReason1. ∀x(P(x) ∨ Q(x))Premise2. P(c) ∨ Q(c) for arbitrary cUniversal instantiation from (1)3. P(c) for arbitrary cSimplification from (2)4. ∀xP(x)Universal generalization from (3)5. Q(c) for arbitrary cSimplification from (2)6. ∀xQ(x)Universal generalization from (5)7.(∀xP(x)) ∨ (∀xQ(x))Disjunction from (4) and (6) (Check all that apply.)
The errors are in step 3 and step 5, because the simplification rule is (P(x) ∧ Q(x)) → P(x).
Translate these system specifications into English, where the predicate S(x, y) is "x is in state y" and where the domain for x and y consists of all systems and all possible states, respectively. ∀x¬S(x, working) (Check all that apply.)
The expression ∀x¬S(x, working) denotes that all systems are not working, which can be stated in two ways: No system is working. Every system is not working.
Express the negations of these propositions using quantifiers, and in English. Let L(x, y) represent x likes the subject y, where x ranges over all the students of our class and y ranges over all the subjects. Express the negation of the proposition "Every student in this class likes mathematics" using quantifiers, and in English.
The given proposition can be expressed as ∀xL(x, mathematics). The negation of ∀xL(x, mathematics) is ∃x¬L(x, mathematics). The expression ∃x¬L(x, mathematics) means that some student in this class does not like mathematics.
Express the negations of these propositions using quantifiers, and in English. Let S(x, y) represent x has seen the object y, where x ranges over all the students of our class and y ranges over all the objects. Express the negation of the proposition "There is a student in this class who has never seen a computer" using quantifiers, and in English.
The given proposition can be expressed as ∃x¬S(x, computer). The negation of ∃x¬S(x, computer) is ∀xS(x, computer). The expression ∀xS(x, computer) means that every student in this class has seen a computer.
Express the negations of these propositions using quantifiers, and in English. Let T(x, c) represent a student x has taken a mathematics course c offered at this school, where x ranges over all the students in the class and c ranges over all the mathematics courses offered in the school. Express the negation of the proposition "There is a student in this class who has taken every mathematics course offered at this school" using quantifiers, and in English.
The given proposition can be expressed as ∃x∀cT(x, c). The negation of ∃x∀cT(x, c) is ∀x∃c¬T(x, c). The expression ∀x∃c¬T(x, c) means that for every student in this class, there is a mathematics course that this student has not taken.
Express the negations of these propositions using quantifiers, and in English. Let P(z, y) be "Room z is in building y," and let Q(x, z) be "Student x has been in room z." Express the negation of the proposition "There is a student in this class who has been in at least one room of every building on campus" using quantifiers, and in English.
The given proposition can be expressed as ∃x∀y∃z(P(z, y) ∧ Q(x, z)). The negation of ∃x∀y∃z(P(z, y) ∧ Q(x, z)) is ∀x∃y∀z(P(z, y) →¬Q(x, z)). The expression ∀x∃y∀z(P(z, y) →¬Q(x, z)) means that for every student, there is a building such that for every room in that building, the student has not been in that room.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let C(x, y) be the predicate that x has climbed y, where x ranges over people and yranges over mountains in the Himalayas. Express the statement "No one has climbed every mountain in the Himalayas" using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English.
The given statement can be expressed as ¬∃x∀y C(x, y). The negation of ¬∃x∀y C(x, y) is ∃x∀y C(x, y). The expression ∃x∀y C(x, y) means that someone has climbed every mountain in the Himalayas.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let S(x, y) mean that student x has solved exercise y, and let B(y, z) mean that exercise y is in section z of the book. Express the statement "No student has solved at least one exercise in every section of this book" using quantifiers.
The given statement can be expressed as ¬∃x∀z∃y(B(y, z) ∧ S(x, y)). The negation of ¬∃x∀z∃y(B(y, z) ∧ S(x, y)) is ∃x∀z∃y(B(y, z) ∧ S(x, y)). The expression ∃x∀z∃y(B(y, z) ∧ S(x, y)) means that some student has solved at least one exercise in every section of this book.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let L(x, y) mean that person x has lost y dollars playing the lottery. Express the statement "No one has lost more than one thousand dollars playing the lottery" using quantifiers.
The given statement can be expressed as ¬∃x∃y(y > 1000 ∧ L(x, y)). The negation of ¬∃x∃y(y > 1000 ∧ L(x, y)) is ∃x∃y(y > 1000 ∧ L(x, y)). The expression ∃x∃y(y > 1000 ∧ L(x, y)) means that someone has lost more than $1000 playing the lottery.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let E(x, y) mean that person x has sent e-mail to person y. Express the statement "No student in this class has sent e-mail to exactly two other students in this class" using quantifiers.
The given statement can be expressed as ¬∃x∃y∃z(y ≠ z ∧ x ≠ y∧ x ≠ z ∧ ∀w(w ≠ x → (E(x, w) ↔ (w = y ∨ w = z)))). The negation of ¬∃x∃y∃z(y ≠ z ∧ x ≠ y ∧ x ≠ z ∧ ∀w(w ≠ x → (E(x, w) ↔ (w = y ∨ w = z)))) is ∃x∃y∃z(y ≠ z ∧ x ≠ y ∧ x ≠ z ∧ ∀w(w ≠ x → (E(x, w) ↔ (w = y ∨ w = z)))). The expression ∃x∃y∃z(y ≠ z ∧ x ≠ y ∧ x ≠ z ∧ ∀w(w ≠ x → (E(x, w) ↔ (w = y ∨ w = z)))) means that some student in this class has sent e-mail to exactly two other students in this class.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let M(x, y, z) be the predicate that x has been in movie z with y, where the domains for x and y are movie actors and for z is movies. Express the statement "Every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon" using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English.
The given statement can be expressed as ∀x((∃zM(x, Kevin Bacon, z)) ∨ (∃y∃z1∃z2(M(x, y, z1) ∧ M(y, Kevin Bacon, z2)))). The negation of ∀x((∃zM(x, Kevin Bacon, z)) ∨ (∃y∃z1∃z2(M(x, y, z1) ∧ M(y, Kevin Bacon, z2)))) is ∃x((∀z¬M(x, Kevin Bacon, z)) ∧ (∀y∀z1∀z2(¬M(x, y, z1) ∨ ¬M(y, Kevin Bacon, z2)))). The expression ∃x((∀z¬M(x, Kevin Bacon, z)) ∧ (∀y∀z1∀z2(¬M(x, y, z1) ∨ ¬M(y, Kevin Bacon, z2)))) means that there is someone who has neither been in a movie with Kevin Bacon nor been in a movie with someone who has been in a movie with Kevin Bacon.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let T(x, y) be the predicate that x has taken y, where x ranges over students in this class and y ranges over mathematics classes at this school. Express the statement "Every student in this class has taken exactly two mathematics classes at this school" using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English.
The given statement can be expressed as ∀x∃y∃z(y ≠ z ∧ T(x, y) ∧ T(x, z) ∧ ∀w(T(x, w) → (w = y ∨ w = z))). The negation of ∀x∃y∃z(y ≠ z ∧ T(x, y) ∧ T(x, z) ∧ ∀w(T(x, w) → (w= y ∨ w = z))) is ∃x∀y∀z(y ≠ z → (¬T(x, y) ∨ ¬T(x, z) ∨ ∃w(T(x, w) ∧ w ≠ y ∧ w ≠ z))). The expression ∃x∀y∀z(y ≠ z → (¬T(x, y) ∨ ¬T(x, z) ∨ ∃w(T(x, w) ∧ w ≠ y ∧ w ≠ z))) means that there is someone in this class for whom no matter which two distinct math courses you consider, these are not the two and only two math courses this person has taken.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let V(x, y) be the predicate that x has visited y, where x ranges over people and yranges over countries. Express the statement "Someone has visited every country in the world except Libya" using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English.
The given statement can be expressed as ∃x∀y(V(x, y) ↔ y ≠ Libya). The negation of ∃x∀y(V(x, y) ↔ y ≠ Libya) is ∀x∃y(V(x, y) ↔ y = Libya). The expression ∀x∃y(V(x, y) ↔ y = Libya) means that for every person, either that person has visited Libya or else that person has failed to visit some country other than Libya.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let S(x, y) mean that student x has solved exercise y. Express the statement "Some student has solved every exercise in this book" using quantifiers.
The given statement can be expressed as ∃x∀yS(x, y). The negation of ∃x∀yS(x, y) is ∀x∃y¬S(x, y). The expression ∀x∃y¬S(x, y) means that for every student in this class, there is some exercise that he or she has not solved.
Express each of these statements using quantifiers. Then, form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. Let C(x, y) mean that person x has chatted with person y. Express the statement "There is a student in this class who has chatted with exactly one other student" using quantifiers.
The given statement can be expressed as ∃x∃y(y ≠ x ∧ ∀z(z ≠ x → (z = y ↔ C(x, z)))). The negation of ∃x∃y(y ≠ x ∧ ∀z(z ≠ x → (z = y ↔ C(x, z)))) is ∀x∀y(y ≠ x → ∃z(z ≠ x ∧ ¬(z = y ↔ C(x, z)))). The expression ∀x∀y(y ≠ x → ∃z(z ≠ x ∧ ¬(z = y ↔ C(x, z)))) means that everybody in this class has either chatted with no one else or has chatted with two or more others.
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃x∃y∀z((x ≠ y) ∧ (W(x, z) ↔ W(y, z)))
There are two different people who have visited exactly the same websites.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃x1∃x2∀z((x1 ≠ x2) ∧ (C(x1, z) ↔ C(x2, z)))
There exist two distinct people enrolled in exactly the same class.
Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃x1∃x2∀z((x1 ≠ x2) ∧ (C(x1, z) → C(x2, z)))
There exist two distinct people, the second of whom is enrolled in every class that the first is enrolled in.
If we let the domain be all animals, and S(x) = "x is a spider", I(x) = " x is an insect", D(x) = "x is a dragonfly", L(x) = "x has six legs", E(x, y ) = "x eats y", then the premises be "All insects have six legs," (∀x (I(x)→ L(x))) "Dragonflies are insects," (∀x (D(x)→I(x))) "Spiders do not have six legs," (∀x (S(x)→¬L(x))) "Spiders eat dragonflies." (∀x, y (S(x) ∧ D(y)) → E(x, y))) Identify the conclusions that can be drawn from the given premises using existential generalization. (Check all that apply.)
There exists a noninsect that eats an insect. There exists a non-six-legged creature that eats a six-legged creature.
Translate these statements into English, where C(x) is "x is a comedian," F(x) is "x is funny," and the domain consists of all people. ∃x(C(x) → F(x))
There exists a person such that if she or he is a comedian, then she or he is funny.
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃y∀z(y ≠ (David Belcher) ∧ (W(David Belcher, z) → W(y, z)))
There is a person besides David Belcher who has visited all the websites that David Belcher has visited.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∃y∀xQ(x, y)
There is a student in your class who has been sent an e-mail message by every student in your class.
Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃∃ y(W(Ashok Puri, y) ∧∧ W(Cindy Yoon, y))
There is a website that both Ashok Puri and Cindy Yoon have visited.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∃x∀yQ(x, y)
There is some student in your class who has sent an e-mail message to every student in your class.
Let Q(x, y) be the statement "x has sent an e-mail message to y," where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. ∃x∃yQ(x, y)
There is some student in your class who has sent an e-mail message to some student in your class.
Let the premises be the statements "All foods that are healthy to eat do not taste good," "Tofu is healthy to eat," "You only eat what tastes good," "You do not eat tofu," and Cheeseburgers are not healthy to eat." Identify the conclusion from the statements "For all x, if x is healthy to eat, then x does not taste good" and "Tofu is healthy to eat."
Tofu does not taste good.
Determine whether the given conditional statements is true or false. If 1 + 1 = 3, then dogs can fly.
True
Determine whether the given conditional statements is true or false. If 1 + 1 = 3, then unicorns exist.
True
Determine whether the given conditional statements is true or false. If 2 + 2 = 4, then 1 + 2 = 3.
True
Consider the following argument. If George does not have eight legs, then he is not a spider. George is a spider. Therefore George has eight legs. Can we conclude that the conclusion is true if the premises are true?
YES
What rule of inference is used in each of these arguments? Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major.
addition
Consider three integers m, n, and p. What kind of proof was used to find that m + p is even?
direct
Let the premises be the statements "I am either dreaming or hallucinating," "I am not dreaming," and "If I am hallucinating, I see elephants running down the road." Identify the rule of inference that is used to arrive at the conclusion "I am hallucinating" from the statements "I am either dreaming or hallucinating," "I am not dreaming." (You must provide an answer before moving to the next part.)
disjunctive syllogism
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. Identify the rule of inference that is used to arrive at the conclusion ∃x(c(x) ∧ w(x)) from the hypothesis w(y) ∧ c(y)).
existential generalization
Let the argument be "Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket." c(x): x is in this class. r(x): x owns a red convertible t(x): x has gotten a speeding ticket. Identify the rule of inference used to arrive at the conclusion ∃x(c(x) ∧ t(x)) from the premises c(Linda) ∧ t(Linda).
existential generalization
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. Identify the rule of inference that is used to arrive at the conclusion ∃x(c(x) ∧ l(x)) from the statement c(y) ∧ l(y).
existential generalization
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. Identify the rule of inference used to arrive at the statement s(y) ∧ c(y) from the statement ∃x(s(x) ∧ c(x)).
existential instantiation
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. Identify the rule of inference that is used to arrive at the statement c(y) ∧ f(y) from the statement ∃x(c(x) ∧ f(x)). (You must provide an answer before moving to the next part.)
existential instantiation
Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? If n is a real number such that n > 1, then n2 > 1. Suppose that n2 > 1. Then n > 1.
fallacy of affirming the conclusion
Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? If n is a real number with n > 2, then n2 > 4. Suppose that n ≤ 2. Then n2 ≤ 4.
fallacy of denying the hypothesis
Suppose that Prolog facts are used to define the predicates mother(M, Y) and father(F, X), which represent that M is the mother of Y and F is the father of X, respectively. Give a Prolog rule to define the predicate grandfather(X, Y), which represents that X is the grandfather of Y. Use a comma to denote "and" and a semicolon to denote "or."
grandfather(X, Y) :- father(X, Z), father(Z, Y); father(X, Z), mother(Z, Y)
Let the argument be "Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year." r(x): x is one of the five roommates listed. d(x): x has taken a course in discrete mathematics. a(x): x can take a course in algorithms. Identify the rule of inference that is used to arrive at the statement r(y) → a(y) from the statements r(y) → d(y) and d(y) → a(y).
hypothetical syllogism
What rule of inference is used in each of these arguments? If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.
hypothetical syllogism
For each of these arguments, determine whether the argument is correct or incorrect along with the reason why the argument form is valid or invalid. Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major.
invalid: fallacy of affirming the conclusions
For each of these arguments, determine whether the argument is correct or incorrect along with the reason why the argument form is valid or invalid. All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit.
invalid: fallacy of denying the hypothesis
If we let the domain be all animals, and S(x) = "x is a spider", I(x) = " x is an insect", D(x) = "x is a dragonfly", L(x) = "x has six legs", E(x, y ) = "x eats y", then the premises be "All insects have six legs," (∀x (I(x)→ L(x))) "Dragonflies are insects," (∀x (D(x)→I(x))) "Spiders do not have six legs," (∀x (S(x)→¬L(x))) "Spiders eat dragonflies." (∀x, y (S(x) ∧ D(y)) → E(x, y))) Identify the rule of inference that is used for the conditional statement "∀x, If x is an insect, then x has six legs" and the statement "Dragonflies are insects" to arrive at the conclusion "Dragonflies have six legs."
modus ponens
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. Identify the rule of inference that is used to arrive at the statement w(y) from the statement s(y) → w(y).
modus ponens
Let the argument be "Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket." c(x): x is in this class. r(x): x owns a red convertible t(x): x has gotten a speeding ticket. Identify the rule of inference that is used to arrive at the conclusion t(Linda) from the premises r(Linda) and r(Linda) → t(Linda).
modus ponens
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. Identify the rule of inference that is used to arrive at the statement l(y) from the statements f(y) → l(y) and f(y).
modus ponens
Let the premises be the statements "All foods that are healthy to eat do not taste good," "Tofu is healthy to eat," "You only eat what tastes good," "You do not eat tofu," and Cheeseburgers are not healthy to eat." Identify the rule of inference that is used to arrive at the conclusion from the statements "For all x, if x is healthy to eat, then x does not taste good" and "Tofu is healthy to eat."
modus ponens
Let the premises be the statements "I am either dreaming or hallucinating," "I am not dreaming," and "If I am hallucinating, I see elephants running down the road." Identify the rule of inference that is used to arrive at the conclusion that I see elephants running down the road from the statements "I am hallucinating," and "If I am hallucinating, I see elephants running down the road."
modus ponens
What rule of inference is used in each of these arguments? If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.
modus ponens
Let the premises be the statements "All foods that are healthy to eat do not taste good," "Tofu is healthy to eat," "You only eat what tastes good," "You do not eat tofu," and Cheeseburgers are not healthy to eat." Identify the rule of inference that is used to derive the conclusion "You do not eat tofu" from the statements "For all x, if x is healthy to eat, then x does not taste good," "Tofu is healthy to eat," and "You only eat what tastes good."
modus tollens
Let the premises be the statements "Every student has an Internet account," "Homer does not have an Internet account," and "Maggie has an Internet account." Identify the rule of inference that is used to arrive at the conclusion that Homer is not a student.
modus tollens
Let the statements "If I play hockey, then I am sore the next day," "I use the whirlpool if I am sore," and "I did not use the whirlpool" be the given premises. Identify the rule of inference that is used to arrive at the conclusion that I am not sore. (You must provide an answer before moving to the next part.)
modus tollens
What rule of inference is used in each of these arguments? If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today
modus tollens
What rule of inference is used in this argument? "No man is an island. Manhattan is an island. Therefore, Manhattan is not a man
modus tollens
Let the argument be "All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners." s(x): x is a movie produced by Sayles. c(x): x is a movie about coal miners. w(x): Movie x is wonderful. Identify the rule of inference that is used to arrive at the statements s(y) and c(y) from the statements s(y) ∧ c(y).
simplification
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. Identify the rule of inference that is used to arrive at the statement c(y) and the statement f(y) from the statement c(y) ∧ f(y).
simplification
What rule of inference is used in each of these arguments? Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.
simplification
Let the argument be "Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year." r(x): x is one of the five roommates listed. d(x): x has taken a course in discrete mathematics. a(x): x can take a course in algorithms. Identify the rule of inference that is used to derive the statements r(y) → d(y) and d(y) → a(y) from the statements ∀x(r(x) → d(x)) and ∀x(d(x) → a(x)). (You must provide an answer before moving to the next part.)
universal instantiation
Let the argument be "Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket." c(x): x is in this class. r(x): x owns a red convertible t(x): x has gotten a speeding ticket. Identify the rule of inference that is used to derive the statement r(Linda) → t(Linda) from the statement ∀x(r(x) → t(x)).
universal instantiation
Let the argument be "There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre." c(x): x is in this class. f(x): x has been to France. l(x): x has visited the Louvre. Identify the rule of inference that is used to arrive at the statement f(y) → l(y) from the statement ∀x(f(x) → l(x)).
universal instantiation
Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? If n is a real number with n > 3, then n2 > 9. Suppose that n2 ≤ 9. Then n ≤ 3.
valid argument using modus tollens
Find a counterexample, to the given universally quantified statements, where the domain consists of all real numbers. ∀x(x2 ≠ x) (Check all that apply.)
x = 0 x = 1
Find a counterexample, to the given universally quantified statements, where the domain consists of all real numbers. ∀x(x2 ≠ 2) (Check all that apply.)
x = √2 x = −√2
Find a counterexample, to the given universally quantified statements, where the domain consists of all real numbers. ∀x(∣∣x∣∣>0)∀xx>0
x=0
Let P(x) be the statement "x > 1." Suppose we initially assign "x := 0" and then execute the statement "if P(x) then x := 1." What is the value of x after the statement is executed? The value of x
x=0
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Let the domain consist of all students in your class. Express the given statement in terms of C(x), D(x), F(x), quantifiers, and logical connectives. For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.
∃xC(x) ∧ ∃xD(x) ∧ ∃xF(x)
Consider the following propositions. Let S(x) denote "x can keep a secret," where the domain is the set of all people. Express the negation of the proposition "No one can keep a secret" using quantifiers, and then express the negation in English.
∃xS(x); Some people can keep a secret.
Let the domain of discourse be the set of all students in your class. If S(x, y) means x can speak the language y, express the statement "There is a student in this class who can speak Hindi" using quantifiers
∃xS(x, Hindi)
Consider the following propositions. Let S(x) denote "x is serious," where the domain is the set of all Swedish movies. Express the negation of the proposition "All Swedish movies are serious" using quantifiers, and then express the negation in English.
∃x¬S(x); Some Swedish movies are not serious.
Let the domain of discourse be the set of all students in your class. If G(x, y) means persons x and y grew up in the same town, express the statement "Some student in this class grew up in the same town as exactly one other student in this class" using quantifiers.
∃x∃y(x ≠ y ∧ G(x, y) ∧ ∀z(G(x, z) → (x = y ∨ x = z)))
Let the domain of discourse be the set of all students in your class. If T(x, y) means x has taken course y and O(y, z) means course y is offered by department z, express the statement "There is a student in this class who has taken every course offered by one of the departments in this school" using quantifiers.
∃x∃z∀y(O(y, z) → T(x, y))
How many rows appear in a truth table for this compound propositions? (p→r)∨(¬s→¬t)∨(¬u→v)(p→r)∨(¬s→¬t)∨(¬u→v) The number of rows needed for the truth table of the compound proposition (p→r)∨(¬s→¬t)∨(¬u→v)(p→r)∨(¬s→¬t)∨(¬u→v) is ____.
64
How many rows appear in a truth table for this compound propositions? (p∨¬t)∧(p∨¬s)(p∨¬t)∧(p∨¬s) The number of rows needed for the truth table of the compound proposition (p∨¬t)∧(p∨¬s)(p∨¬t)∧(p∨¬s) is ____.
8
Translate these statements into English, where C(x) is "x is a comedian," F(x) is "x is funny," and the domain consists of all people. ∀x(C(x) ∧ F(x))
Every person is a funny comedian
Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. Lunch includes soup or salad.
Exclusive or
Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. Publish or perish.
Exclusive or
Find the dual of the given compound proposition. The dual of the compound proposition p ∧ (q ∨ (r ∧ T)) is p ∨ (q ∧ (r ∨ T)).
FALSE
For each of these statements, find a domain for which the statement is true or false. The statement "No two different people have the same grandmother" is false Correctin the domain "all residents of the United States."
FALSE
Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(lemon)? The truth value of P(lemon) is
FALSE
Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(true)? The truth value is
FALSE
Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(Detroit, Michigan). The truth value of Q(Detroit, Michigan) is
FALSE
Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(Massachusetts, Boston). The truth value of Q(Massachusetts, Boston) is
FALSE
Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(New York, New York). The truth value of Q(New York, New York) i
FALSE
Determine whether the given sentence is an inclusive or, or an exclusive or, is intended. To enter the country you need a passport or a voter registration card.
Inclusive or
Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∃x((x ≥ 0) ∧ P(x)) without using quantifiers, instead using only negations, disjunctions, or conjunctions. Click on the choice which corresponds to a correct solution
P(1) ∨ P(3) ∨ P(5)
Express the given sentence in terms of P(x), Q(x), quantifiers, and logical connectives. Click and drag the appropriate word, symbol, or phrase into the most appropriate blank.
P(x) "x can speak Russian Q(x) x knows the computer language C++
Suppose that the domain of the propositional function P(x) consists of −5, −3, −1, 1, 3, and 5. Express ∀x((x ≠ 1) → P(x)) without using quantifiers, instead using only negations, disjunctions, or conjunctions. Click on the choice or choices which correspond to a correct solution
P(−5) ∧ P(−3) ∧ P(−1) ∧ P(3) ∧ P(5)
Let Q(x) be the statement "x + 1 > 2x," where the domain consists all integers. Identify the quantifications that have the truth value "true." (Check all that apply.)
Q(0) Q(-1) ∃ x¬Q(x) ∃∃ xQ(x)
Determine whether the given compound propositions is satisfiable. (p ∨ q ∨ r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (q ∨ ¬r ∨ s) ∧ (¬p ∨ r ∨ s) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬r) ∧ (¬p ∨ ¬q ∨ s) ∧ (¬p ∨ ¬r ∨ ¬s) The compound proposition (p ∨ q ∨ r) ∧ (p ∨ ¬q ∨ ¬s) ∧ (q ∨ ¬r ∨ s) ∧ (¬p ∨ r ∨ s) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬r) ∧ (¬p ∨ ¬q ∨ s) ∧ (¬p ∨ ¬r ∨ ¬s) is
Satisfiable
Determine whether the given compound propositions is satisfiable. (¬p ∨ ¬q ∨ r) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬s) ∧ (¬p ∨ ¬r ∨ ¬s) ∧ (p ∨ q ∨ ¬r) ∧ (p ∨ ¬r ∨ ¬s) The compound proposition (¬p ∨ ¬q ∨ r) ∧ (¬p ∨ q ∨ ¬s) ∧ (p ∨ ¬q ∨ ¬s) ∧ (¬p ∨ ¬r ∨ ¬s) ∧ (p ∨ q ∨ ¬r) ∧ (p ∨ ¬r ∨ ¬s) is
Satisfiable
Translate these statements into English, where C(x) is "x is a comedian," F(x) is "x is funny," and the domain consists of all people. ∃x(C(x) ∧ F(x))
Some comedians are funny.
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Let the domain consist of all students in your class. The quantification for the below statement is expressed in terms of C(x), D(x), F(x), quantifiers, and logical connectives as ∀x(C(x) ∨ D(x) ∨ F(x)). All students in your class have a cat, a dog, or a ferret.
TRUE
Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(false)? The truth value is
TRUE
Let P(x) be the statement "the word x contains the letter a." What is the truth value of P(orange)? The truth value of P(orange) is
TRUE
Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(Denver, Colorado). The truth value of Q(Denver, Colorado) is
TRUE
The statement "Every two people have the same mother" is false Incorrectin the domain "George W. Bush and Jeb Bush."
TRUE
State the converse, contrapositive, and inverse of this conditional statement. Click and drag the named related conditionals to their corresponding statements of the conditional statement "If it snows today, I will ski tomorrow."
The converse is "If I am to ski tomorrow, it must snow today." The inverse is "If it does not snow today, then I will not ski tomorrow." The contrapositive is "If I do not ski tomorrow, then it will not have snowed today."
State the converse, contrapositive, and inverse of this conditional statement. Click and drag the named related conditionals to their corresponding statements of the conditional statement "I come to class whenever there is going to be a quiz."
The converse is "If I come to class, then there will be a quiz." The contrapositive is "If I don't come to class, then there won't be a quiz." The inverse is "If there is not going to be a quiz, then I don't come to class."
Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement ∃x(R(x) Λ H(x)) into English. (Check all that apply.)
The existential quantifier states that the conditional statement applies to at least one animal. Thus, "There exists an animal such that if it is a rabbit, then it hops" or "Some hopping animals are rabbits."
Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement ∃x(R(x) → H(x)) into English. (Check all that apply.)
The existential quantifier states that the conditional statement applies to at least one animal. Thus, "There exists an animal such that if it is a rabbit, then it hops."
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. F(x) be the expression "x can swim," and G(x) be "x can catch fish," and the domain of discourse is pigs. The statement is "There exists a pig that can swim and catch fish."
The expression for the statement is ∃∃ x (F(x) ∧∧ G(x)), and its negation is ∀∀ x¬(F(x) ∧∧ G(x)) and the sentence is "No pig can both swim and catch fish."
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. F(x) be the expression "x can climb," and the domain of discourse is koalas. The statement is "Every koala can climb."
The expression is ∀∀ x F(x), its negation is ∃∃ x¬F(x) and the sentence is "There is a koala that cannot climb."
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. F(x) be the expression "x has fleas," and the domain of discourse is dogs. The statement is "All dogs have fleas."
The expression is ∀∀ x F(x), its negation is ∃∃ x¬F(x), and the sentence is "There is a dog that does not have fleas."
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. F(x) be the expression "x can speak French," and the domain of discourse is monkeys. The statement is "No monkey can speak French."
The expression is ∀∀ x¬F(x), its negation is ∃∃ x F(x) and the sentence is "There is a monkey that can speak French."
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. F(x) be the expression "x can add," and the domain of discourse is horses. The statement is "There is a horse that can add."
The expression is ∃∃ x F(x), its negation is ∀∀ x¬F(x) and the sentence is "No horse can add."
Identify the correct steps involved in proving p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent. (Check all that apply.
The first statement p ↔ q is true if and only if p and q have the same truth value. If both p and q are true, then (p ∧ q) is true and (¬p ∧ ¬q) is false. This implies (p ∧ q) ∨ (¬p ∧ ¬q) is true. Similarly, if both p and q are false, then (p ∧ q) is false and (¬p ∧ ¬q) is true. This implies (p ∧ q) ∨ (¬p ∧ ¬q) is true. Thus, p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) have same truth value; hence, they are logically equivalent.
State the converse, contrapositive, and inverse of this conditional statement. Click and drag the named related conditionals to their corresponding statements of the conditional statement "A positive integer is a prime only if it has no divisors other than 1 and itself."
The inverse is "If a positive integer is not prime, then it has a divisor other than 1 and itself." The contrapositive is "If a positive integer has a divisor other than 1 and itself, then it is not prime." The converse is "A positive integer is a prime if it has no divisors other than 1 and itself."
Let P(x) be "x is perfect" let F(x) be "x is your friend"; and let the domain be all people. Click and drag the quantifications to their corresponding sentences.
The statement "No one is perfect." can expressed as ∀x¬P(x) and ¬∃xP(x). The statement "Not everyone is perfect." can be expressed as ¬∀xP(x). The statement "All your friends are perfect." can be expressed as ∀x(F(x) → P(x)). The statement "At least one of your friends is perfect." can be expressed as ∃x(F(x) ∧ P(x)). The statement "Everyone is your friend and is perfect." can be expressed as ∀x(F(x) ∧ P(x)) and ∀x(F(x)) ∧ ∀xP(x)). The statement "Not everybody is your friend or someone is not perfect." can be expressed as (¬∀xF(x)) ∨ (∃x¬P(x)).
Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement "∀x(R(x) → H(x))" into English. (Check all that apply.)
The statenment ∀x(R(x) → H(x)) can be expressed as "If an animal is a rabbit, then that animal hops" and "Every rabbit hops."
Fill the provided truth tables and check whether the given conditional statements are a tautology or contradiction or contingency. Is the conditional statement ¬(p → q)→ ¬q tautology?
Yes it is tautology
Let P(x) be the statement "x > 1." Suppose we initially assign "x := 1" and then execute the statement "if P(x) then x := 1". What is the value of x after the the statement is executed? The value of x =
x=1
Let P(x) be the statement "x > 1." Suppose we initially assign "x := 2" and then execute the statement "if P(x) then x := 1". What is the value of x after the statement is executed? The value of x =
x=1
Consider the set of all propositions. If T(x) means that x is a tautology and C(x) means that x is a contradiction, express each of these statements using logical operators, predicates, and quantifiers. The negation of a contradiction is a tautology.
∀x(C(x) → T(¬x))
Consider the set of all propositions. If T(x) means that x is a tautology and C(x) means that x is a contradiction, express each of these statements using logical operators, predicates, and quantifiers. The conjunction of two tautologies is a tautology.
∀x∀y((T(x) ∧ T(y)) → T(x ∧ y))
dentify the statements that have the truth value "true" if the domain of each variable consists of all real numbers. (Check all that apply.)
∃ x(x2 = 2) ∀ x(x^2 + 2 ≥ 1)
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Let the domain consist of all students in your class. Express the statement below in terms of C(x), D(x), F(x), quantifiers, and logical connectives. A student in your class has a cat, a dog, and a ferret.
∃x(C(x) ∧ D(x) ∧ F(x))
et N(x) be the statement "x has visited North Dakota," where the domain consists of the students in your school. Click and drag any of the given English statements and place them next to the quantifications provided. A quantification may be matched to more than one English statement provided.
∃xN(x) : There is a student at your school who has visited North Dakota.At least one student at your school has visited North Dakota.Some students at your school have visited North Dakota. ∀xN(x) : Every student at your school has visited North Dakota.All students at your school have visited North Dakota. ¬∃xN(x) : It is not the case that a student at your school has visited North Dakota. No student at your school has visited North Dakota. ∃x¬N(x): There is a student at your school who has not visited North Dakota.At least one student at your school has not visited North Dakota.Some students at your school have not visited North Dakota. ∀x¬N(x) : All students at your school have not visited North Dakota.Each student at your school has not visited North Dakota. ¬∀xN(x) : It is not the case that each student at your school has visited North Dakota.
Consider the set of all propositions. If T(x) means that x is a tautology and C(x) means that x is a contradiction, express each of these statements using logical operators, predicates, and quantifiers. Some propositions are tautologies.
∃xT(x)
Consider the set of all propositions. If T(x) means that x is a tautology and C(x) means that x is a contradiction, express each of these statements using logical operators, predicates, and quantifiers. The disjunction of two contingencies can be a tautology.
∃x∃y(¬T(x) ∧ ¬C(x) ∧ ¬T(y) ∧ ¬C(y) ∧ T(x ∨ y))
