Prove that [4x)A(x)]' → (Vx)[A(x)]' is valid. We must prove the implication in each direction. a. [ax)4(x)]' <→ (Vx)[A(x)]' The hypothesis alone gives us little to work with, so we introduce a (somewhat surprising) temporary hypothesis. A proof sequence is 1. [Ex)A(x)]' 2. A(x) 3. (Ex)A(x) hyp temporary hyp 2, eg 4. A(x) → (Ax)A(x) temporary hyp discharged 5. [A(x)]' 1, 4, mt 6. (Vx)[A(x)]' 5, ug b. (Vx)[4(x)]' → [x)A(x)]' This proof also requires a temporary hypothesis. It is even more surprising than case (a) because we assume the exact opposite of the conclusion we are trying to reach. 1. (Vx)[A(x)]' 2. (Ex)4(x) hyp temporary hyp 3. A(a) 2, ei 4. [A(a)]' 1, ui 5. [(Vx)[A(x)]'I' 6. (Ax)A(x) → [(Vx)[A(x)]]' temporary hyp discharged 3, 4, inc 7. [((Vx)[A(x)])]' 1, dn 8. [Ex)4(x)]' 6, 7, mt

Prove that [4x)A(x)]' → (Vx)[A(x)]' is valid. We must prove the implication in each direction. a. [ax)4(x)]' <→ (Vx)[A(x)]' The hypothesis alone gives us little to work with, so we introduce a (somewhat surprising) temporary hypothesis. A proof sequence is 1. [Ex)A(x)]' 2. A(x) 3. (Ex)A(x) hyp temporary hyp 2, eg 4. A(x) → (Ax)A(x) temporary hyp discharged 5. [A(x)]' 1, 4, mt 6. (Vx)[A(x)]' 5, ug b. (Vx)[4(x)]' → [x)A(x)]' This proof also requires a temporary hypothesis. It is even more surprising than case (a) because we assume the exact opposite of the conclusion we are trying to reach. 1. (Vx)[A(x)]' 2. (Ex)4(x) hyp temporary hyp 3. A(a) 2, ei 4. [A(a)]' 1, ui 5. [(Vx)[A(x)]'I' 6. (Ax)A(x) → [(Vx)[A(x)]]' temporary hyp discharged 3, 4, inc 7. [((Vx)[A(x)])]' 1, dn 8. [Ex)4(x)]' 6, 7, mt

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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