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Chapter 3: Problem 26

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

Short Answer

Expert verified
The truth table for \( \sim q \wedge p \) is: \n p q |\(\sim q\) | \(\sim q \wedge p\)\n T T | F | F\n T F | T | T\n F T | F | F\n F F | T | F

Step by step solution

01

Construct the Base Truth Table

First, create a truth table containing 'p' and 'q' with all possible true(T) or false(F) combinations. There should be four rows to account for all possible combinations.
02

Calculate \(\sim q\)

Our next step is to calculate the negation of 'q' (\(\sim q\)). The negation simply reverses the truth value, so 'T' becomes 'F' and vice versa. A third column is added to your truth table to account for this.
03

Calculate \(\sim q \wedge p\)

Now, it is time to bring in the 'and' operation on '\(\sim q\)' and 'p'. Remember that for an 'and' operation to be true both operands must be true. A fourth column is added to your truth table in which you put true or false depending on whether both '\(\sim q\)' and 'p' are true or not.

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Key Concepts

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

Logical Negation
Logical negation is a fundamental concept in logic, often symbolized by the "not" operator \( \sim \). It is a unary operation, which means it only works on a single proposition at a time. The purpose of logical negation is simple: it reverses the truth value of the proposition it is applied to.

For example, if we have a proposition \( q \) that is "True" (T), applying negation (\( \sim q \)) changes it to "False" (F). Similarly, if \( q \) is originally "False" (F), then \( \sim q \) becomes "True" (T).
  • If \( q \) is True, \( \sim q \) is False.
  • If \( q \) is False, \( \sim q \) is True.

This operation is crucial in constructing more complex logical expressions and is one of the building blocks for forming logical statements in truth tables.
Logical Conjunction
Logical conjunction, often represented by the "and" operator \( \wedge \), is a binary operation that combines two propositions. In the context of truth tables, it is crucial for determining when two conditions are simultaneously true.

The rule for logical conjunction is straightforward: the combined statement is true only if both individual statements are true. If at least one of the statements is false, the conjunction results in false.
  • If both \( p \) and \( q \) are True, then \( p \wedge q \) is True.
  • If either \( p \) or \( q \) is False, then \( p \wedge q \) is False.

This operation is used extensively in forming complex logical expressions and is key when analyzing conditions where multiple criteria must all be satisfied.
Constructing Truth Tables
Constructing truth tables is a systematic way to explore the truth values of logical expressions. It involves breaking down complex expressions into their simpler components. To build a truth table, follow these steps:

  • Identify the variables: Determine all the propositions involved, like \( p \) and \( q \).

  • List all combinations: Write out all possible combinations of truth values for the variables. If there are two variables, like \( p \) and \( q \), there will be four combinations: TT, TF, FT, FF.

  • Add columns for each operation: Include separate columns for each logical operation involved. For example, negation \( \sim q \) and conjunction \( \sim q \wedge p \).

  • Fill in truth values: Calculate the truth values for each operation according to their rules, filling them into the appropriate columns.

Constructing a truth table allows one to visualize and understand complex logical expressions, step by step.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All clocks keep time accurately. All time-measuring devices keep time accurately. Therefore, all clocks are time-measuring devices.

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.

Write an original argument in words for the contrapositive reasoning form.

Write an example of an argument with two quantified premises that is invalid but that has a true conclusion.

Use Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is an athlete. Therefore, Savion Glover is a dancer.
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