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Chapter 1: Problem 26

Construct a truth table for each proposition. $$p \wedge(q \wedge r)$$

Short Answer

Expert verified
The completed truth table for the proposition \(p \wedge (q \wedge r)\) is: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F |

Step by step solution

01

List all possible combinations of truth values for \(p\), \(q\), and \(r\)

We have 3 propositions: \(p\), \(q\), and \(r\). Each proposition can either be true (T) or false (F). Therefore, there are \(2^3 = 8\) possible combinations of truth values for these 3 propositions. List all 8 combinations in a table: | \(p\) | \(q\) | \(r\) | |-----|-----|-----| | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F |
02

Determine the truth value of \((q \wedge r)\) for each row

For each row in the table, determine the truth value of \((q \wedge r)\) using the truth table definition of conjunction (AND): \(\wedge\) is true if and only if both propositions are true. Add a column for \((q \wedge r)\) in the truth table: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | |-----|-----|-----|---------------| | T | T | T | T | | T | T | F | F | | T | F | T | F | | T | F | F | F | | F | T | T | T | | F | T | F | F | | F | F | T | F | | F | F | F | F |
03

Determine the truth value of \(p \wedge (q \wedge r)\) for each row

For each row in the table, determine the truth value of the compound proposition \(p \wedge (q \wedge r)\) using the truth table definition of conjunction (AND): \(\wedge\) is true if and only if both propositions are true. Add a column for \(p \wedge (q \wedge r)\) in the truth table: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F | Now, the completed truth table for the proposition \(p \wedge (q \wedge r)\) is as follows: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F |

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Propositions
Propositions are fundamental components in logic that help in forming functional statements. A proposition is basically a declarative sentence that contains a true or false value but not both at the same time. For instance, statements like "The sky is blue." are propositions because it is either true or false based on observation.

  • Propositions must be distinct and bear a truth value in logical exercises.

  • They are represented in logic as variables, such as \( p, q, \) and \( r \) in our example.

  • These variables can be used in various configurations to develop complex expressions, which can further be evaluated with truth tables.

Whenever you encounter logical expressions or problems, identifying and understanding individual propositions is the first crucial step.
Conjunction
The concept of conjunction is a logical operation that combines two propositions and evaluates to true only when both propositions are true. This operation is commonly known as "AND," and it plays a crucial role in constructing compound logical statements.

  • The symbol for conjunction is \( \wedge \), so \( p \wedge q \) reads as "\( p \) AND \( q \)."

  • This operation is intuitive: it reflects situations where multiple conditions must be satisfied simultaneously.

  • In the context of sets, conjunctions reflect the intersection of conditions where all involved propositions agree.

In practical terms, when evaluating conjunction in a truth table, you mark the outcome as true only if every proposition considered in the "AND" expression is true. This strict requirement ensures the precision of logical conclusions.
Logical Operators
Understanding logical operators is essential when dealing with logical expressions. They are special symbols or words used to connect propositions to form more complex logical statements. Each operator has a distinct function and truth condition.

  • For example, conjunction (\( \wedge \)) is one of the primary logical operators used for combining statements.

  • Other common operators include disjunction (\( \vee \)), which represents "OR," and negation (\( eg \)), which represents "NOT."

  • Logical operators are the building blocks of logic used to analyze statements and their relations.

Each operator follows specific rules that set how the truth values of individual propositions influence the truth value of the compound expression. Mastering these operators is crucial for creating and analyzing logical statements accurately in various settings.

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Most popular questions from this chapter

Refer to Example 1.32 and are based on Smullyan's What is the name of this book? A and \(\mathrm{B}\) are inhabitants of the island. What are they if A says each of the following? A says, "All of us are knaves," and B says, "Exactly one of us is a knave." What is C?

Three gentlemen - Mr. Blue, Mr. Gray, and Mr. White-have shirts and ties that are blue, gray, and white, but not necessarily in that order. No person's clothing has the same color as his last name. Mr. Blue's tie has the same color as Mr. Gray's shirt. What color is Mr. White's shirt? (Mathematics Teacher, 1986)

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \vee q \equiv q \vee p$$

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \wedge \sim p \equiv f$$

"How is it, Professor Whipple," asked a curious student, "that someone as notoriously absentminded as you are manages to remember his telephone number?" "Quite simple, young man" replied the professor. "I simply keep in mind that it is the only seven-digit number such that the number obtained by reversing its digits is a factor of the number." What is Professor Whipple's telephone number? (A. J. Friedland, 1970 )
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