2. Let p, q, r be propositions. Based on given truth values of individual propositions or compound propositions decide the missing truth value(s), if possible. Pick the correct answer (circle or underline, or use a color). No explanation is necessary. a) If p = T Aq =F Ar =T then (p → ng) O (p ^ r) = -. TRUTH FALSE cannot be decided b) If p =T nq = F Ar = F then (p Oq) → (q → r) = ... TRUTH FALSE cannot be decided

2. Let p, q, r be propositions. Based on given truth values of individual propositions or compound propositions decide the missing truth value(s), if possible. Pick the correct answer (circle or underline, or use a color). No explanation is necessary. a) If p = T Aq =F Ar =T then (p → ng) O (p ^ r) = -. TRUTH FALSE cannot be decided b) If p =T nq = F Ar = F then (p Oq) → (q → r) = ... TRUTH FALSE cannot be decided

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

Problem 1PE

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2) (L2) Prove using laws of logic that the conditional proposition (p ∧ q) → r is equivalent to (p ∧ ¬ r) →¬ q. 3) (L3) Show that the converse of a conditional proposition p: q → r is equivalent to the inverse of proposition p using a truth table. 4.1) (L4) Show whether ((p ∧ (p→q)) ↔ ¬p) is a tautology or not. Use a truth table and be specific about which row(s)/column(s) of the truth table justify your answer. 4.2) (L4) Give truth values for the propositional variables that cause the two expressions to have different truth values. For example, given p ∨ q and p ⊕ q, the correct answer would be p = q = T, because when p and q are both true, p ∨ q is true but p ⊕ q is false. Note that there may be more than one correct answer. r ∧ (p ∨ q) (r ∧ p) ∨ q
Assume that ∀x∃yP(x,y) is true and that the universe id true for x and y is non empty. Which of the following is true ? ∃x∀yP(x,y) and ∃y∃xP(x,y) . Prove your answer either using regular proof in the natural language or by giving a formal inference . In that latter case u may need to use existential rules of inference .
QUESTION 2 Consider the following propositions : p: "I finish my Work" q: "I play handball" r: "The sun is shining" s: "The humidity is low" The symbolic of the following sentence "I finish my work and I play handball whenever the humidity is low" is : p logical and q rightwards arrow s None of all the proposed answers s logical and p logical or q s rightwards arrow p logical and q not s rightwards arrow not p logical or not q
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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
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Consider the following arguments, in each case determine whether the argument is valid, i.e., is it an instanceof an argument form in deductive logic in which the rule of inference is truth-preserving?(a) 1. p2. (p ∨ s) (b) 1. ¬(r ∨ s)2. ((r ∨ s) ∨ p)3. p (c) 1. (¬r ∨ ¬s)2. ((p ∨ q) → (r ∧ s))3. ¬(p ∨ q) (d) 1. (p → ¬¬p)2. (¬¬¬p ∧ ¬p)3. ¬p4. (¬p ∨ q)
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In the domain of all integers, let number (x ) denote the statement “x is a number”, and composite(x ) denote the statement “x is a composite.” Formalize the following sentences and indicate their corresponding formal type.1. “Some numbers are not composite.”2. “No numbers are prime.”3. “Some numbers are not prime.”4. “All numbers are prime.” ========================================================================= For subject S and predicate P, we can express the Type A categorical proposition as ∀ s : S | s ∈ P1. Prove formally that negating A is logically equivalent to obtaining O (and vice versa).2. Prove formally that negating E is logically equivalent to obtaining I (and vice versa).