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Chapter 3: Problem 32

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

Short Answer

Expert verified
The truth table is as follows:| p | q | \( \sim p \) | \( \sim q \) | \( p \vee \sim q \) | \( \sim p \wedge (p \vee \sim q) \) ||---|---|------|------|---------------|------------------------|| T | T | F | F | T | F || T | F | F | T | T | F || F | T | T | F | F | F || F | F | T | T | T | T |

Step by step solution

01

Define all possible combinations for p and q

There are 4 different possibilities for the boolean variables \( p \) and \( q \) as they can either be True (T) or False (F). These are: TT, TF, FT, FF.
02

Determine the NOT operation

The NOT operation (\( \sim \)) inverts the value of the variable that follows it. Therefore, \( \sim p \) and \( \sim q \) will take the opposite value of \( p \) and \( q \) for each of the combinations.
03

Determine the OR operation

Calculate the value of the OR operation (\( \vee \)) for each combination, which involves \( p \) and \( \sim q \). In an OR operation, the result is True if any or both of the inputs are True, else it's False.
04

Determine the AND operation

Finally, calculate the value of the AND operation (\( \wedge \)) for each combination, which involves \( \sim p \) and the result of the OR operation (\( p \vee \sim q \)). In an AND operation, the result is True if both the inputs are True, else it's False.

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Key Concepts

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

Boolean Variables
Boolean variables are the foundation of logic-based systems. They represent values that can be either True or False. In our example, we use the boolean variables \( p \) and \( q \). They can take two possible values each, leading to a combination of outcomes:
  • True-True (TT)
  • True-False (TF)
  • False-True (FT)
  • False-False (FF)
These combinations are the basis for constructing a truth table, helping us evaluate logical statements across different scenarios.
Logical Operations
Logical operations are used to manipulate boolean variables and expressions. They include operations like AND, OR, and NOT, which perform basic but powerful functions. In logical computations, these operations transform inputs into meaningful outputs, allowing us to form complex logical expressions.
Negation
Negation, denoted by \( \sim \), is a fundamental logical operation that inverts the value of a boolean variable. If the variable is True, negation makes it False, and vice versa.
For example, given the statement \( \sim p \), if \( p \) is True, \( \sim p \) will be False.
In our scenario, understanding negation is crucial as it helps reverse the state of a variable, influencing the outcome of combined logical operations.
Disjunction
Disjunction is represented by the logical operator \( \vee \), known as OR. It results in True if at least one of the operands is True. If both operands are False, the result is False.
  • If both \( p \) or \( \sim q \) (not\( q \)) are True, the disjunction \( p \vee \sim q \) yields True.
  • It turns False only if both \( p \) is False and \( q \) is True.
Disjunction allows the inclusion of possibilities, making it an essential part of logical operations.
Conjunction
Conjunction is represented by the operator \( \wedge \), often referred to as AND. It returns True only if both input conditions are True; if either input is False, the operation yields False.
  • For the operation \( \sim p \wedge (p \vee \sim q) \), conjunction checks if both \( \sim p \) and the result of \( p \vee \sim q \) are True.
  • This narrow condition ensures more restrictive outcomes, which can be beneficial for defining precise logical conditions.
Conjunction is crucial when precise and strict conditions need to be met in logical expressions.

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Most popular questions from this chapter

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. In order to avoid a long truth table and instead use a standard form of an argument, I tested the validity of Limbaugh's argument using the following representations: \(p\) : Gore really believes that the automobile is a threat to the planet. \(q:\) He does not travel in a gas guzzler.

No animals that eat meat are vegetarians. No cat is a vegetarian. Felix is a cat. Therefore, ... a. Felix is a vegetarian. b. Felix is not a vegetarian. c. Felix eats meat. d. All animals that do not eat meat are vegetarians.

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. You exercise or you do not feel energized. I do not exercise. Therefore, ...

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used Euler diagrams to determine that an argument is valid, but when I reverse one of the premises and the conclusion, this new argument is invalid.

In Symbolic Logic, Lewis Carroll presents the following argument: Babies are illogical. (All babies are illogical persons.) Illogical persons are despised. (All illogical persons are despised persons.) Nobody is despised who can manage a crocodile. (No persons who can manage crocodiles are despised persons.) Therefore, babies cannot manage crocodiles. Use an Euler diagram to determine whether the argument is valid or invalid.
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