3. Find the negations of each of the following statements:(a) Every math major is smart and attractive.(b) Not all smart attractive people are math majors.(c) Ervy [P(x) V ¬Q (y)]

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Answered: 3. Find the negations of each of the…

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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Which of the following is a logical equivalence in first-order logic? A. ¬(∀x)P(x) ≡ (∃x)¬P(x) B. (∀x)(∃y)P(x, y) ≡ (∃y)(∀x)P(x, y) C. (∀x)(P(x) → Q(x)) ≡ (∃x)(P(x) ∧ Q(x)) D. (∀x)(P(x) ∧ Q(x)) ≡ (∀x)(P(x) ∨ Q(x))
1. For the following statements, write in symbolic language with quantifiers, negate the statement in symbolic language, determine if the statement is true or false and justify your answer with a proof or counterexample. (a) There is a natural number greater than two that is not the sum of two primes. (b) There is a prime that is one more than a multiple of 4 and is also a sum of two squares. (c) For any natural number n, there is a natural numbers that is the product of n with itself. (d) There is a natural number s such that for any natural number n, s is the product of n with itself. (e) If the product of a rational number x and a non-zero real number y is rational, then x is zero or y is rational. (f) If a and b are integers then there are integers m and n such that a = m+n and b=m-n.
21. Let the following predicates be given. The domain is all computer science classes. I(r) = "x is interesting." U(x) = "x is useful." H(x, y) = “x is harder than y." M(x, y) = “x has more students than y." (a) Write the following statements in predicate logic. i. All interesting CS classes are useful. ii. There are some useful CS classes that are not interesting. iii. Every interesting CS class has more students than any non- interesting CS class. (b) Write the following predicate logic statement in everyday English. Don't just give a word-for-word translation; your sentence should make sense. (3)[I(x) A (Vy)(H(x,y)→ M(y, x))] (c) Formally negate the statement from part (b). Simplify your negation so that no quantifier lies within the scope of a negation. State which derivation rules you are using. (d) Give a translation of your negated statement in everyday English.
4. Let P be "It is humid" and let Q be "sum of 9 and 11 is 21". Give a simple verbal sentence which describes each of the following statements: (a) ¬Q, (b) P V Q, (c) (¬P) ^ Q, (d) (¬P) V (–¬Q).
3. Let us consider the propositional language where: A = "Peter does his homework" B = "Burt does his homework" C = "Andy does his homework" D = "Dexter comes to the party" Write the statement for the following symbols. (d) C → (~D → ~A) (e) (~D → C) A (D → ~B) (f) A A~BAC → ~D) a ( A a B → (D → C)
If p is true, q is true, and r is false, determine whether each of the following compound statements aretrue or false”(a) p ∧ q(b) ∼ (q ∨ r)(c) r → (p ∧ q)(d) (p ∧ r) ∨ (p ∧ q → r)
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3. Find the negations of each of the following statements:.. (a) Every math major is smart and attractive. (b) Not all smart attractive people are math majors. (c) Ervy [P(x) ✓ ¬Q (y)]