Got a b and c done need help with d and e please show work **Step 1****Disclaimer:** Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts for you. To get the remaining sub-parts solved, please repost the complete question and mention the sub-parts to be solved.The logical symbol "∨" is used to denote "OR". The logical symbol "↓" is used to denote "NOT". Any two propositions are said to be logically equivalent if the truth value in each row of the truth table is the same for both propositions.**(a)**It is provided that P↓Q is true if both P, Q are false and false in all other scenarios. Use this data to make the truth table of P↓Q as follows.| P | Q | P↓Q ||---|---|-----|| T | T | F || T | F | F || F | T | F || F | F | T |Make the truth table of ¬P∨Q as follows.| P | Q | P∨Q | ¬P∨Q ||---|---|-----|------|| T | T | T | F || T | F | T | F || F | T | T | T || F | F | F | T |Comparing the last columns of both truth tables, it can be concluded that the expressions: P↓Q and ¬P∨Q are logically equivalent.**(b)**Make the truth table of Q↓P as follows.| P | Q | Q↓P ||---|---|-----|| T | T | F || T | F | F || F | T | F || F | F | T |Comparing the last columns of both truth tables, it can be concluded that the expressions: P↓Q and Q↓P are logically equivalent. Hence, ↓ is commutative.**(c)**Make the truth table of P↓Q↓R and P↓(Q↓R) as follows.| P | Q | R | P↓Q | P↓Q↓R | Q↓R | P↓(Q↓R) ||---|---|---|-----|-------|-----|---------|| T | T | T | F | F | F | F 1. We define a logical connective ↓ as follows: \( P \downarrow Q \) is true when both \( P \) and \( Q \) are false, and it is false otherwise. (We read \( P \downarrow Q \) as “P nor Q”).(a) Write a truth table for \( P \downarrow Q \) and check that \( P \downarrow Q \) is logically equivalent to \(\neg(P \lor Q)\).(b) Check that \( P \downarrow Q \equiv Q \downarrow P \). That is, ↓ is commutative.(c) Show that \( (P \downarrow Q) \downarrow R \) and \( P \downarrow (Q \downarrow R) \) are logically inequivalent. That is, ↓ is not associative.(d) Show that the logical connectives \(\neg\), \(\land\), and \(\lor\) can each be expressed entirely in terms of ↓, without using any other logical connectives. Specifically, prove the following: i. \(\neg P \equiv (P \downarrow P)\). ii. \(P \land Q \equiv (P \downarrow P) \downarrow (Q \downarrow Q)\). iii. \(P \lor Q \equiv (P \downarrow Q) \downarrow (P \downarrow Q)\).(e) Prove that the logical connective \(\Rightarrow\) can be expressed entirely in terms of ↓. That is, show that the sentence \( P \Rightarrow Q \) is logically equivalent to a sentence involving ↓ and no other logical connectives.

Part D) (i) P ¬P T F F T as mentioned that P↓Q is true only when both are false , and in…

1. We define a logical connective as follows: P↓ Q is true when both P and Q are false, and it is false otherwise. (We read P Q as "P nor Q"). (a) Write a truth table for P Q and check that P Q is logically equivalent to ¬(PVQ). (b) Check that P↓ Q = Q ↓ P. That is, is commutative. (c) Show that (P ↓ Q) ↓ R and P↓ (QR) are logically inequivalent. That is, is not associative. (d) Show that the logical connectives, A, and V can each be expressed entirely in terms of ↓, without using any other logical connectives. Specifically, prove the following: i. ¬P = (PP). PAQ=(PP) ↓ (Q ↓ Q). iii. PVQ = (P↓ Q) ↓ (P ↓ Q). (e) Prove that the logical connective ⇒ can be expressed entirely in terms of ↓. That is, show that the sentence P⇒ Q is logically equivalent to a sentence involving ↓ and no other logical connectives.

1. We define a logical connective as follows: P↓ Q is true when both P and Q are false, and it is false otherwise. (We read P Q as "P nor Q"). (a) Write a truth table for P Q and check that P Q is logically equivalent to ¬(PVQ). (b) Check that P↓ Q = Q ↓ P. That is, is commutative. (c) Show that (P ↓ Q) ↓ R and P↓ (QR) are logically inequivalent. That is, is not associative. (d) Show that the logical connectives, A, and V can each be expressed entirely in terms of ↓, without using any other logical connectives. Specifically, prove the following: i. ¬P = (PP). PAQ=(PP) ↓ (Q ↓ Q). iii. PVQ = (P↓ Q) ↓ (P ↓ Q). (e) Prove that the logical connective ⇒ can be expressed entirely in terms of ↓. That is, show that the sentence P⇒ Q is logically equivalent to a sentence involving ↓ and no other logical connectives.

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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