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Chapter 3: Problem 37

Construct a truth table for the given statement. \(p \wedge(\sim q \vee r)\)

Short Answer

Expert verified
The truth table for the statement \(p \wedge(\sim q \vee r)\) is obtained by determining the truth values for each component of the statement based on the logical connectors and the truth values for the individual propositions. The complete truth table will include all possible combinations of truth values for p, q, and r, and the corresponding truth values for the compound statement.

Step by step solution

01

Understand the Symbols

Recognize that the given statement \(p \wedge(\sim q \vee r)\) involves three different propositions: p, q, and r. Likewise, it includes the logical connectors 'and' denoted by \(\wedge\), 'not' denoted by \(\sim\), and 'or' denoted by \(\vee\).
02

Construct the Basic Truth Table

Begin by constructing the truth table for the basics. This includes the three propositions: p, q, and r. Each of these can take on either of two truth values: true (T) or false (F). So, with three variables, there will be \(2^3 = 8\) rows for all the possible combinations of truth values.
03

Determine the Truth Value for \(\sim q\)

Next, calculate the truth values for \(\sim q\). This is the negation of proposition q, meaning its truth value will be the opposite of whatever truth value q has. If q is true, then \(\sim q\) is false, and vice versa.
04

Determine the Truth Value for \((\sim q \vee r)\)

Now determine \((\sim q \vee r)\). This is the 'or' operation between \(\sim q\) and r. According to 'or' operation rule, the statement is true if either \(\sim q\) or r or both are true.
05

Determine the Whole Statement's Truth Value

Finally, get the truth values for the whole statement, \(p \wedge(\sim q \vee r)\). This is the 'and' operation between p and \((\sim q \vee r)\). According the 'and' operation rule, the statement is true if both p and \((\sim q \vee r)\) are true.

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Key Concepts

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

Logical Connectors
In logic, we use specific symbols to represent the relationships between propositions. These symbols are known as logical connectors. They help us form complex statements from simpler ones. A proposition is a statement that can either be true or false, but not both.
The main logical connectors include:
  • AND (\(\wedge\)): It is used when both propositions need to be true for the entire statement to be true. Example: "It is sunny AND warm" means it must be both sunny and warm for the overall statement to be true.
  • OR (\(\vee\)): It means the statement is true if at least one proposition is true. For example, "It is sunny OR rainy" is true if it's sunny, rainy, or both.
  • NOT (\(\sim\)): This flips the truth value of a proposition. If something is true, "NOT" makes it false and vice versa.
By understanding these connectors, you can interpret and construct logical statements and truth tables.
Negation
Negation is a fundamental concept in logic that involves the logical connector "NOT." It reverses the truth value of a given proposition. For example, if a proposition \( q \) is "The sky is blue," its negation \( \sim q \) would be "The sky is NOT blue."
The rules of negation are simple:
  • If the original statement is true, the negation is false.
  • If the original statement is false, the negation is true.
Negation plays a critical role in logical expressions and truth tables. In our exercise, when you negate \( q \), denoted by \( \sim q \), it changes all true instances of \( q \) to false, and all false instances to true.
Disjunction
Disjunction involves the use of the logical "OR" connector, represented by \( \vee \). This operation combines two propositions and results in a true value if at least one of the propositions is true. This flexibility makes it widely useful in logical expressions.
The truth rules for disjunction are:
  • The statement is false only if both propositions are false.
  • The statement is true if either or both propositions are true.
In our specific example with \( \sim q \vee r \), the statement will hold true if either \( \sim q \) is true, \( r \) is true, or both are true. The concept of disjunction is pivotal in understanding how multiple conditions can result in a true outcome.
Conjunction
Conjunction uses the logical "AND" connector, denoted by \( \wedge \). This operation mandates that both propositions involved must be true for the overall expression to be true. It is often used when all conditions in a scenario must be satisfied.
The truth rules for conjunction are:
  • The statement is false if either proposition is false.
  • It is true only when both propositions are true.
For the logical statement \( p \wedge (\sim q \vee r) \) from our exercise, conjunction is used to combine \( p \) with the outcome of \( \sim q \vee r \). Both \( p \) and the result of \( \sim q \vee r \) must be true for the entire expression to hold true. This requirement for simultaneous truth emphasizes the restrictive nature of conjunction in logical statements.

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If some journalists learn about the invasion, the newspapers will print the news. If the newspapers print the news, the invasion will not be a secret. The invasion was a secret. \(\therefore\) No journalists learned about the invasion.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders are insects. Therefore, no spiders have six legs.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders have six legs. Therefore, no spiders are insects.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of 3 . Therefore, 8 is not a multiple of 6 .

Write a valid argument on one of the following questions. If you can, write valid arguments on both sides. a. Should the death penalty be abolished? b. Should Roe \(v\). Wade be overturned? c. Are online classes a good idea? d. Should recreational marijuana be legalized? e. Should grades be abolished? f. Should the Electoral College be replaced with a popular vote?
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