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Chapter 3: Problem 72

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

Short Answer

Expert verified
The truth value for \(\sim p \leftrightarrow(\sim q \wedge r)\) is false.

Step by step solution

01

Determine individual negations

Identify the truth values of the negations of each statement. Given \(p\) is false, \(\sim p\) (not p) is true. Given \(q\) is true, \(\sim q\) (not q) is false. Given \(r\) is false, its truth value remains the same.
02

Evaluate the conjunction

Evaluate the truth value of the conjunction (\(\wedge\)), \(\sim q \wedge r\). In a conjunction, both elements must be true for the entire conjunction to be true. As we determined in step 1, \(\sim q\) is false and \(r\) is false, hence, \(\sim q \wedge r\) is false.
03

Evaluate the biconditional

Evaluate the truth value for the biconditional (\(\leftrightarrow\)), \(\sim p \leftrightarrow(\sim q \wedge r)\). A biconditional is true if and only if both components have the same truth value. From step 1, \(\sim p\) is true. From step 2, \(\sim q \wedge r\) is false. Since these two components do not have the same truth value, \(\sim p \leftrightarrow(\sim q \wedge r)\) is false.

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Key Concepts

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

Understanding Truth Values
Truth values are fundamental to logical reasoning. They are basically the "True" or "False" possible states that a statement or proposition can take. Think of truth values as the digital world of logic: everything boils down to zeroes and ones. Whenever you encounter a logical proposition, determining its truth value is crucial.
  • If the proposition is correct, it is assigned a truth value of True.
  • If the proposition is incorrect, it is given a truth value of False.
In logical exercises, like the one from the original exercise, you start by determining the truth values of each proposition. These values help us systematically analyze more complex logical structures.
The Concept of Negation
Negation is like saying "it's the opposite day" for logical statements. When you negate a proposition, you flip its truth value:
  • If the original statement is True, its negation is False.
  • If the original statement is False, its negation is True.
In the original exercise, when we negated the statement \( p \) 鈥� originally False 鈥� it turned into True. Negation is an essential tool in logic, allowing us to question assumptions and explore scenarios from opposing perspectives.
Decoding Conjunction
Conjunction, symbolized by (\(\wedge\)), works like an 'AND' between statements. For a conjunctive statement to be True, both elements must be True as well:
  • True AND True = True
  • True AND False = False
  • False AND True = False
  • False AND False = False
In the exercise, the conjunction \( \sim q \wedge r \) resulted in False, because both \(\sim q\) and \( r \) needed to be True, but neither was. Logical conjunctions are critical in scenarios where multiple conditions must be met simultaneously.
Understanding Biconditional
A biconditional statement (\(\leftrightarrow\)) reflects a two-way street in logic: both sides must match in truth value for the statement to be True:
  • True if both sides are True.
  • True if both sides are False.
  • False if one side is True and the other is False.
The original exercise's biconditional \(\sim p \leftrightarrow (\sim q \wedge r)\) was False because \(\sim p\) was True while \(\sim q \wedge r\) was False. When understanding biconditionals, imagine an agreement where both parties must comply for success. If one party defaults, the agreement fails, much like how the truth values operate in logical biconditionals.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an artist. Therefore, Sean Penn is an actor.

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.

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 a truth table to determine whether the symbolic form of the argument is valid or invalid. \(p \rightarrow q\) \(\frac{q \wedge r}{\therefore p \vee r}\)

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I am a full-time student, I cannot work. If I cannot work, I cannot afford a rental apartment costing more than \(\$ 500\) per month. Therefore, ...
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