#88. Determine whether these pairs of statements arelogically equivalent. Justify your answer.a) Vx(P(x)Q(x)) and VxP(x)VxQ(x)b) 3x(P(x) V Q(x)) and 3xP(x) v 3xQ(x).->You may copy the sign , A, V, →, +, =, 3!, 3,1.

Equivalence using quantifiers. ∀(P(x)) →Q(x)) and ∀xP(x) →∀xQ(x) This is not actually logically…

Answered: B. Determine whether these pairs of…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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1): Select the law which shows that the two propositions are logically equivalent. ¬((w∨p)∧(¬q∧¬w)) ¬(w∨p)∨¬(¬q∧¬w) A): DeMorgan’s law B): Distributive law C): Associative law D): Complement law 2): Select the proposition that is a tautology. (p∧q)→¬p (p∨q)→p (p∧q)↔p 4 . (p∧q)→p A): 4 B): 1 C): 2 D): 3
Let P(x) and Q(x) be relations, with x a variable from a given domain of discourse U. The Axiom of the Hilbert deductive system for first-order logic applied below is "Axiom 1,2,3,4,5" ? ⊢ ∀x P(x) → (∃x Q(x) → ∀x P(x)) Consider the steps given below of a proof in the Hilbert deductive system and determine the reason that justifies each. ∀x A(x) ⊢ ∀x A(x) "?"∀x A(x) ⊢ ∀x A(x) → A(a) " ?"∀x A(x) ⊢ A(a) " ?"
Consider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’ throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to n atoms.(d) State the number of rows in a truth table for proving the extension
Consider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to natoms.(d) State the number of rows in a truth table for proving the extension.
Complete the axiomatization by using and add a rule of universal generalization (∀1∀1) ∀x A→A(y/x) ∀x A→A(y/x),provided yy is free for xx in AA (∀2∀2) ∀x(A→B) → (A→∀x B) ∀x(A→B) → (A→∀x B),provided xx does not occur free in A
Please answer the following question in depth with full detail. Consider the 8-puzzle that we discussed in class. Suppose we define a new heuristic function h3 which is the average of h1 and h2, and another heuristic function h4 which is the sum of h1 and h2. That is, for every state s ∈ S: h3(s) =h1(s) + h2(s) 2 h4(s) =h1(s) + h2(s) where h1 and h2 are defined as “the number of misplaced tiles”, and “the sum of the distances of the tiles from their goal positions”, respectively. Are h3 and h4 admissible? If admissible, compare their dominance with respect to h1 and h2, if not, provide a counterexample, i.e. a puzzle configuration where dominance does not hold.
Determine whether the following proposition is a tautology: (¬p∨¬(r⟶q))⟷(p⟶(¬q∧r)) can i get a non handwriting answer so it would be easy to copy please
We have shown how to use truth tables to determine if two formulas are truth-functionally equivalent.If two formulas F and G are truth-functionally equivalent we introduce another symbol ↔, aptly called thebiconditional. Here is the truth table for the biconditional.p q (p ↔ q)1 1 11 0 00 1 00 0 1Now we shall say that F and G are truth-functionally equivalent if (F ↔ G) is a tautology.There are other properties of two formulas that we are usually interested in besides truth-functionalequivalence. One of these properties is when two formulas are mutually exclusive. We say two formulas Fand G are mutually exclusive if (F ∧ G) is contradictory (unsatisfiable).Now using truth tables determine whether the following formulas are truth-functionally equivalent or mutuallyexclusive.(a) p and ¬p (b) p and ¬¬p (c) ¬(p ∧ ¬q) and (p → q) (d) (¬p ∨ q) and (p → q) (e) ¬(¬p ∨ ¬q) and (p ∧ q) (f) ¬(¬p ∧ ¬q) and (p ∨ q)
Show that each of these conditional statements is a tautology. Please show each step and the laws you use. 1) ¬p→(p→q) 2) (p∧q)→(p→q) Example: (p∧q)→p ≡¬(p∧q)∨p [Identity of implies] ≡(¬p∨¬q)∨p [De Morgan’s law] ≡(¬p∨p)∨¬q [Associative law] ≡T∨¬q [Negation law] ≡T [Domination law]
Let P(x) be the statement “x can speak Russian” and let Q(x) be the statement “x knows C++”. Express each of these statements in terms P(x), Q(x), quantifiers and logical connectives. The domain for quantifiers consists of all students at Mines. There is a student at Mines who can speak Russian and who knows C++. Every student at your school either can speak Russian or knows C++.
1. With the aid of Truth tables, determine which of the following logical statements (or expressions)is a Tautology, a Contradiction, or a Contingency.(a) (p ↔ q) ↔ ((p → q) ∧ (q → p))(b) (p → (q → r)) ↔ (p ∧ q → r)(c) (p → r ∨ q)⊕(q ∧ r)(d) p ∨ q ∧ (¬r → ¬p ∧ ¬q) 2. With reference to the following logical equivalences, determine whether each of them is valid orinvalid. If it is invalid then give a counterexample (e.g. based on a Truth-Value assignment).If it is valid then give an algebraic proof using ONLY logical equivalences from Table 6 inSection 1.3 of course textbook, and the rule for conditional: p → q ≡ ¬p ∨ q.(a) p → (q → r) ≡ (p → q) → r(b) (p → r) ∨ (q → r) ≡ (p ∧ q) → r(c) (¬p → (q ∧ ¬q)) ≡ p(d) p ∧ q → r ≡ (¬p ∧ ¬q) → ¬r 3. Express the following logical statements in terms of the Contrapositive, Converse, and Inverse,respectively.(a) You are an adult provided that you are above 18 years of age.(b) It is daylight whenever the sun is shining.(c) 3 × 2 = 6 only if 2 × 3 = 6.
ASAP. will upvote and leave positive remark Q1. Simplify the following Boolean expressions using K-maps: F(x,y,z) = x’y’ + yz + x’yz’ F(x,y,z) = x’y + yz’ + y’z’ F(w,x,y,z) = x’z + w’xy’ + w(x’y + xy’) Q2. Find all the prime implicants for the following Boolean functions, and determine which are essential: F(w,x,y,z) = ∑(0,2,4,5,6,7,8,10,13,15) F(A,B,C,D) = ∑(0,2,3,5,7,8,10,11,14,15)

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B. Determine whether these pairs of statements are logically equivalent. Justify your answer. a) Vx(P(x)→ Q(x)) and VxP(x)→ VxQ(x) b) 3x(P(x) V Q(x)) and ExP(x) V IxQ(x).