ii. Let F(x,y) be the open sentence "x can fool y" where the domain ofdiscourse is the set of all people in the world. Express each of thesestatements using logical quantifiers.c. Everybody can fool somebody.e. Everyone can be fooled by somebody.f. There is somebody who can be fooled by everybody.

Answered: Image /qna-images/answer/d45e5046-dba1-4ff8-adb8-d728c545e288.jpg

Answered: ii. Let F(x,y) be the open sentence "x…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let Q(x) be the statement "x + 1 > 2x." If the universe of discourse consists of all integers, which of these predicates will be true? A) vx ¬Q(x) Vx Q(x) c) Q(-1) D) Q(1)
Exercise 1. Put the following sentences into symbolic logic using single variable predi- cates and quantifiers and then negate them. Define your domain and the predicates. (a) No real number squared is π. • Domain: • Predicate: Statement in symbolic form: Negation in symbolic form: ● Negation in English: . Which is true? The statement or its negation? Justify your answer. (b) Not all even numbers are divisible by 4. • Domain: • Predicate: Statement in symbolic form: Negation in symbolic form: • Negation in English: . Which is true? The statement or its negation? Justify your answer.
3. Let S(x) be the statement “x is a student," P(x) be the statement “x is a professor," and Q(x, y) be the statement “x asked a question to y." If the domain of x and y consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y), quantifiers, and logical operators. Example: Some student has asked every professor a question. Answer. 3x(S(x) A Vy(P(V) → Q(x,y))) a. Every professor has been asked a question by some student. b. There is a student who has never asked a question to a professor. c. There is a professor who has been asked a question by exactly one student. d. There are two different students who have asked each other a question.
3. Let S(x) be the statement “x is a student," P(x) be the statement “x is a professor," and Q(x, y) be the statement “x asked a question to y." If the domain of x and y consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y), quantifiers, and logical operators. Example: Some student has asked every professor a question. Answer: ax(S(x) A Vy(P(y)→ Q(x, y))) a. Every professor has been asked a question by some student. b. There is a professor who has never been asked a question by any student. c. There is a student who has asked a question to exactly one professor. d. There are two different students who have asked each other a question.
2. Given a predicate logic sentence: E: if (for all x)p(x, f(y)) then p(a, f(y)) Show that & is valid.
4. 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. Then determine the truth value of each statement. Give your argument. a) aXVy(x + y = y) b) VxVy(((x > 0) A (y 0))
let P and Q be predicates on some domain of discourse D and define p = Vx (P(x)) → Vx (Q(x)) q = Vx ( P(x) → Q(x) ). Check all statements that must be true. O If q is true, then p is true. None of the other choices. If p is true, then q is true.
5. Rewrite the following proposition as unambiguous English sentences. The relevant predicates are defined as follows: Again, let’s let P be a set of all people a. A(x) means “x is teaching the AI course" b. T(x) means “x is taking AI course" c. F(x) means “x has a twitter account." d. C(x) means “x likes to read." For example, the proposition 3x {P[A(x)^ C(x)]} could be translated as "There is a person who is teaching and likes to read" P [T(x) → (F(x) C(x))] ?
discrete structure
a. Write the set T={ x | x ER, x²= 4 or x² = 25} in list form. b. Let A = {f, t, r}and B = {2, 5,} List the elements of the following sets. i. Power(A ) = ii. Bx A = c. What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers . 1. P(x) is “|x| = 25, " 2. Q(x) is “x² = 144,"
2. Mark each statement True or False. Justify each answer. (a) The symbol “3" means "there exist several." (b) If a variable is used in the antecedent of an implication without being quantified, then the universal quantifier is assumed to apply. (c) The order in which quantifiers are used affects the truth value.
discrete structure

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