Let \[$ P \$\] : The Day Is Tuesday. Let \[$ Q \$\] : The Day Is Thursday. Let \[$ R \$\] : She Comes To Work.Which Represents The Statement The Day Is Tuesday Or Thursday If And Only If She Comes To Work?A. \[$ P = (q

by ADMIN 223 views

Introduction

In the realm of mathematics, particularly in the field of logic, conditional statements play a crucial role in expressing complex relationships between variables. In this article, we will delve into the world of logical expressions and explore how to represent conditional statements using mathematical notation. We will examine a specific problem involving the day of the week and a person's work schedule, and determine which logical expression represents the statement "The day is Tuesday or Thursday if and only if she comes to work."

Understanding Conditional Statements

A conditional statement is a statement that is dependent on the truth value of another statement. It is typically represented in the form "if p, then q," where p is the antecedent and q is the consequent. The conditional statement can be true or false, depending on the truth values of p and q.

Logical Operators

In logic, there are several operators that are used to combine conditional statements. The most common operators are:

  • Conjunction (AND): represented by ∧ (or * in some notation)
  • Disjunction (OR): represented by ∨ (or + in some notation)
  • Negation (NOT): represented by ¬ (or ~ in some notation)
  • Implication (IF-THEN): represented by → (or ⇒ in some notation)

The Problem

Let's examine the problem presented in the introduction:

  • Let p: The day is Tuesday.
  • Let q: The day is Thursday.
  • Let r: She comes to work.

We need to determine which logical expression represents the statement "The day is Tuesday or Thursday if and only if she comes to work."

Logical Expressions

There are several logical expressions that can be used to represent the given statement. Let's examine each option:

Option A: p ∨ q → r

This expression states that if the day is Tuesday or Thursday, then she comes to work. This is a correct representation of the statement, as it implies that if the day is either Tuesday or Thursday, then she must come to work.

Option B: r → (p ∨ q)

This expression states that if she comes to work, then the day is Tuesday or Thursday. This is not a correct representation of the statement, as it implies that if she comes to work, then the day must be either Tuesday or Thursday, which is not necessarily true.

Option C: p ∨ q → (r ∧ ¬r)

This expression states that if the day is Tuesday or Thursday, then she comes to work and she does not come to work. This is not a correct representation of the statement, as it implies that if the day is either Tuesday or Thursday, then she must both come to work and not come to work, which is a logical contradiction.

Option D: (p ∨ q) → r

This expression states that if the day is Tuesday or Thursday, then she comes to work. This is a correct representation of the statement, as it implies that if the day is either Tuesday or Thursday, then she must come to work.

Option E: r → (p ∨ q)

This expression states that if she comes to work, then the day is Tuesday or Thursday. This is not a correct representation of the statement, as it implies that if she comes to work, then the day must be either Tuesday or Thursday, which is not necessarily true.

Conclusion

In conclusion, the correct logical expression that represents the statement "The day is Tuesday or Thursday if and only if she comes to work" is:

  • Option A: p ∨ q → r
  • Option D: (p ∨ q) → r

Both of these expressions correctly represent the statement, as they imply that if the day is either Tuesday or Thursday, then she must come to work.

Final Thoughts

In this article, we have explored the world of logical expressions and conditional statements. We have examined a specific problem involving the day of the week and a person's work schedule, and determined which logical expression represents the statement "The day is Tuesday or Thursday if and only if she comes to work." We have seen that the correct logical expression is p ∨ q → r or (p ∨ q) → r, which implies that if the day is either Tuesday or Thursday, then she must come to work.

References

  • [1] "Logic" by Ian Chiswell and Wilfrid Hodges
  • [2] "A First Course in Logic" by Wilfrid Hodges
  • [3] "Mathematical Logic" by Elliott Mendelson

Glossary

  • Antecedent: The first part of a conditional statement, which is the condition that must be met.
  • Consequent: The second part of a conditional statement, which is the result of the condition being met.
  • Conjunction: A logical operator that combines two or more statements with a logical AND.
  • Disjunction: A logical operator that combines two or more statements with a logical OR.
  • Implication: A logical operator that represents a conditional statement.
  • Negation: A logical operator that represents the opposite of a statement.
    Logical Expressions and Conditional Statements: A Comprehensive Guide ===========================================================

Q&A: Logical Expressions and Conditional Statements

In the previous article, we explored the world of logical expressions and conditional statements. We examined a specific problem involving the day of the week and a person's work schedule, and determined which logical expression represents the statement "The day is Tuesday or Thursday if and only if she comes to work." In this article, we will answer some frequently asked questions about logical expressions and conditional statements.

Q: What is the difference between a conditional statement and a logical expression?

A: A conditional statement is a statement that is dependent on the truth value of another statement. It is typically represented in the form "if p, then q," where p is the antecedent and q is the consequent. A logical expression, on the other hand, is a combination of one or more statements using logical operators.

Q: What are the most common logical operators?

A: The most common logical operators are:

  • Conjunction (AND): represented by ∧ (or * in some notation)
  • Disjunction (OR): represented by ∨ (or + in some notation)
  • Negation (NOT): represented by ¬ (or ~ in some notation)
  • Implication (IF-THEN): represented by → (or ⇒ in some notation)

Q: How do I determine the truth value of a conditional statement?

A: To determine the truth value of a conditional statement, you need to evaluate the truth values of the antecedent and the consequent. If the antecedent is true and the consequent is true, then the conditional statement is true. If the antecedent is false or the consequent is false, then the conditional statement is false.

Q: What is the difference between a biconditional statement and a conditional statement?

A: A biconditional statement is a statement that is true if and only if both the antecedent and the consequent are true. It is typically represented in the form "p if and only if q." A conditional statement, on the other hand, is a statement that is true if the antecedent is true and the consequent is true.

Q: How do I represent a biconditional statement using logical operators?

A: To represent a biconditional statement using logical operators, you can use the following expression:

p ⇔ q = (p → q) ∧ (q → p)

This expression states that p if and only if q is true if and only if both p → q and q → p are true.

Q: What is the difference between a tautology and a contradiction?

A: A tautology is a statement that is always true, regardless of the truth values of the variables. A contradiction, on the other hand, is a statement that is always false, regardless of the truth values of the variables.

Q: How do I determine if a statement is a tautology or a contradiction?

A: To determine if a statement is a tautology or a contradiction, you need to evaluate the truth values of the statement for all possible combinations of truth values of the variables. If the statement is true for all possible combinations of truth values, then it is a tautology. If the statement is false for all possible combinations of truth values, then it is a contradiction.

Q: What is the difference between a satisfiable statement and an unsatisfiable statement?

A: A satisfiable statement is a statement that is true for at least one combination of truth values of the variables. An unsatisfiable statement, on the other hand, is a statement that is false for all possible combinations of truth values of the variables.

Q: How do I determine if a statement is satisfiable or unsatisfiable?

A: To determine if a statement is satisfiable or unsatisfiable, you need to evaluate the truth values of the statement for all possible combinations of truth values of the variables. If the statement is true for at least one combination of truth values, then it is satisfiable. If the statement is false for all possible combinations of truth values, then it is unsatisfiable.

Conclusion

In this article, we have answered some frequently asked questions about logical expressions and conditional statements. We have explored the world of logical expressions and conditional statements, and have determined which logical expression represents the statement "The day is Tuesday or Thursday if and only if she comes to work." We have seen that the correct logical expression is p ∨ q → r or (p ∨ q) → r, which implies that if the day is either Tuesday or Thursday, then she must come to work.

Final Thoughts

In conclusion, logical expressions and conditional statements are fundamental concepts in mathematics and computer science. Understanding these concepts is essential for solving problems and making decisions in a variety of fields. We hope that this article has provided a comprehensive guide to logical expressions and conditional statements, and has helped you to better understand these important concepts.

References

  • [1] "Logic" by Ian Chiswell and Wilfrid Hodges
  • [2] "A First Course in Logic" by Wilfrid Hodges
  • [3] "Mathematical Logic" by Elliott Mendelson

Glossary

  • Antecedent: The first part of a conditional statement, which is the condition that must be met.
  • Consequent: The second part of a conditional statement, which is the result of the condition being met.
  • Conjunction: A logical operator that combines two or more statements with a logical AND.
  • Disjunction: A logical operator that combines two or more statements with a logical OR.
  • Implication: A logical operator that represents a conditional statement.
  • Negation: A logical operator that represents the opposite of a statement.
  • Tautology: A statement that is always true, regardless of the truth values of the variables.
  • Contradiction: A statement that is always false, regardless of the truth values of the variables.
  • Satisfiable statement: A statement that is true for at least one combination of truth values of the variables.
  • Unsatisfiable statement: A statement that is false for all possible combinations of truth values of the variables.