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Chapter 3: Problem 53

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(p \wedge(q \vee r)\)

Short Answer

Expert verified
The truth value of the statement \(p \wedge (q \vee r)\) when \(p\) is false, \(q\) is true, and \(r\) is false is false.

Step by step solution

01

Evaluate \(q \vee r\)

Evaluate the operation within the brackets first according to BODMAS rules. As per the given condition, \(q\) is true and \(r\) is false. The OR (\(\vee\)) operation takes the value true if either of its operands is true. Therefore, \(q \vee r = true \vee false = true\).
02

Evaluate \(p \wedge (q \vee r)\)

Now, evaluate \(p \wedge (q \vee r)\) using the value of \(q \vee r\) and the given value of \(p\). The AND (\(\wedge\)) operation takes the value true only if both of its operands are true. In this case, \(p = false\) and \(q \vee r = true\), so \(p \wedge (q \vee r) = false \wedge true = false\).

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Key Concepts

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

Logical Conjunction
Logical conjunction, commonly represented by the symbol \( \wedge \) and known as the \( AND \) operator, is a fundamental concept in logical operations. It connects two statements (operands) in such a manner that the combined statement is true only if both of the operands are true.
For example, in the provided exercise, the conjunction is used between \( p \) and the result of \( q \vee r \). The truth value of \( p \) is false, and despite the truth value of \( q \vee r \) being true, the entire expression \( p \wedge (q \vee r) \) yields a false result because \( p \) is false, and both operands need to be true for a conjunction to be true.
  • If \( p \) is true and \( q \) is true, then \( p \wedge q \) is true.
  • If either \( p \) or \( q \) is false, then \( p \wedge q \) is false.
Logical Disjunction
Logical disjunction signifies the \( OR \) operation in logic, symbolized by \( \vee \) and it defines a statement that is true if at least one of the operands is true.
The exercise makes use of this principle when it evaluates \( q \vee r \) inside the parentheses. Since \( q \) is true and \( r \) is false, the disjunction \( true \vee false \) becomes true because at least one operand鈥擻( q \)鈥攊s true.
  • If \( p \) is true or \( q \) is true (or both), then \( p \vee q \) is true.
  • If both \( p \) and \( q \) are false, then \( p \vee q \) is false.

Understanding disjunction is critical for analyzing complex logical statements and constructing truth tables in logic.
BODMAS Rules
BODMAS is an acronym representing an order of operations used to solve mathematical expressions: Brackets, Orders (powers and square roots, etc.), Division and Multiplication, Addition and Subtraction.
This rule dictates that operations inside brackets are to be performed first, as seen in the exercise \( p \wedge (q \vee r) \). Here, \( (q \vee r) \) indicates that the operation within the brackets, the logical disjunction \( q \vee r \), must be resolved before applying the logical conjunction \( p \wedge 鈥� \) to the result.
Carefully applying BODMAS ensures the correct order of operations in mathematics and logic and prevents errors in computation of complex expressions.
Logical Operands
In logic, operands are the elements that operators act upon. In the context of the exercise, \( p \) and \( q \) are instances of logical operands.
Each operand has a truth value: true or false. When logical operators like conjunction \( \wedge \) and disjunction \( \vee \) are applied to these values, they create new logical statements.
The truth value of an operand can change the outcome of a logical expression significantly. For example, altering the truth value of \( p \) from false to true would change the final output of the expression \( p \wedge (q \vee r) \) from false to true, showcasing the critical role of operands in logical expressions.

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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 I tell you I cheated, I'm miserable. If I don't tell you I cheated, I'm miserable. \(\therefore\) I'm miserable.

Based on the following argument by conservative radio talk show host Rush Limbaugh and directed at former vice president Al Gore. You would think that if \(\mathrm{Al}\) Gore and company believe so passionately in their environmental crusading that [sic] they would first put these ideas to work in their own lives, right? ... Al Gore thinks the automobile is one of the greatest threats to the planet, but he sure as heck still travels in one of them - a gas guzzler too. (See, I Told You So, p. 168) Limbaugh's passage can be expressed in the form of an argument: If Gore really believed that the automobile were a threat to the planet, he would not travel in a gas guzzler. Gore does travel in a gas guzzler. Therefore, Gore does not really believe that the automobile is a threat to the planet. Use Limbaugh's argument to determine whether each statement makes sense or does not make sense, and explain your reasoning. I think Limbaugh is a fanatic and all his arguments are invalid.

This is an excerpt from a 1967 speech in the U.S. House of Representatives by Representative Adam Clayton Powell: He who is without sin should cast the first stone. There is no one here who does not have a skeleton in his closet. I know, and I know them by name. Powell's argument can be expressed as follows: No sinner is one who should cast the first stone. All people here are sinners. Therefore, no person here is one who should cast the first stone. Use an Euler diagram to determine whether the argument is valid or invalid.

Use Euler diagrams to determine whether each argument is valid or invalid. All thefts are immoral acts. \(\underline{\text { Some thefts are justifiable. }}\) Therefore, some immoral acts are justifiable.

Use Euler diagrams to determine whether each argument is valid or invalid. All dogs have fleas. Some dogs have rabies. Therefore, all dogs with rabies have fleas.
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