For every of 4 statements prove its correctness. **Table 8: Logical Equivalences Involving Biconditional Statements**The table presents logical equivalences for biconditional statements using propositional logic symbols. These equivalences demonstrate different ways to express the biconditional relationship between two propositions, \( p \) and \( q \).1. \( p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p) \) - This states that the biconditional \( p \leftrightarrow q \) is equivalent to both \( p \) implying \( q \) and \( q \) implying \( p \).2. \( p \leftrightarrow q \equiv \neg p \leftrightarrow \neg q \) - This equivalence shows that flipping the truth values of both \( p \) and \( q \) does not alter the truth of the biconditional.3. \( p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q) \) - Here, the biconditional is expressed as the disjunction of both propositions being true or both being false.4. \( \neg (p \leftrightarrow q) \equiv p \leftrightarrow \neg q \) - This indicates that negating a biconditional is equivalent to making one of the propositions the negation in the biconditional.These equivalences are fundamental in understanding logical reasoning and are widely used in mathematical proofs and logical arguments.

A) (p<->q) ≡ ((p->q)∧(q->p)) (p<->q) P q p<->q T T T T F F F T…

Answered: p>q = (p→q)^(q → P) p>q =¬p+¬q p+q =…

Engineering

Computer Science

p>q = (p→q)^(q → P) p>q =¬p+¬q p+q = (p^g)v(¬p^-9) G(p+q) = p+¬9

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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Concept explainers

Heuristic System

The term heuristic is derived from the Greek word heurisko, which means “I find, discover”. Heuristics can be mental shortcuts or approaches designed from previous experiences. The main aim of the heuristic approach is to find a good enough solution to solve the problem. Test and mistake, rule of thumb, or an educated estimate are all examples of heuristics. These methods are sufficient for achieving an immediate, brief aim. When a heuristic is used in practice, it works as predicted. It can, however, produce systematic errors. Trial and error is an important heuristic that may be used in a variety of situations, from cross-linking nuts and bolts to determining the values of given input variables in mathematical puzzles. Additional hypotheses, visual illustrations, simplification, and forward or backward reasoning are all common heuristics in mathematics.

Question

For every of 4 statements prove its correctness.

**Table 8: Logical Equivalences Involving Biconditional Statements**

The table presents logical equivalences for biconditional statements using propositional logic symbols. These equivalences demonstrate different ways to express the biconditional relationship between two propositions, \( p \) and \( q \).

1. \( p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p) \)

- This states that the biconditional \( p \leftrightarrow q \) is equivalent to both \( p \) implying \( q \) and \( q \) implying \( p \).

2. \( p \leftrightarrow q \equiv \neg p \leftrightarrow \neg q \)

- This equivalence shows that flipping the truth values of both \( p \) and \( q \) does not alter the truth of the biconditional.

3. \( p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q) \)

- Here, the biconditional is expressed as the disjunction of both propositions being true or both being false.

4. \( \neg (p \leftrightarrow q) \equiv p \leftrightarrow \neg q \)

- This indicates that negating a biconditional is equivalent to making one of the propositions the negation in the biconditional.

These equivalences are fundamental in understanding logical reasoning and are widely used in mathematical proofs and logical arguments.

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