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Chapter 3: Problem 22

Construct a truth table for the given statement. \((p \leftrightarrow q) \rightarrow q\)

Short Answer

Expert verified
The truth table for the given logic statement \((p \leftrightarrow q) \rightarrow q\) is created with four scenarios based on the possible truth values of the variables p and q. The truth table shows the result of the biconditional \(p \leftrightarrow q\) and the result of the entire statement \((p \leftrightarrow q) \rightarrow q\).

Step by step solution

01

Understand the Logical Connectives

The logical connectives used in this statement are the biconditional (\(\leftrightarrow\)) and the implies (\(\rightarrow\)). The biconditional, also known as the if and only if, is true when both values are the same and false when they are different. The implies is true only when a true value implies another true value, it's false when a true value implies a false value.
02

Determine Variables

There are two variables present in this logic statement: p and q. Therefore, four scenarios are possible from the combinations of True and False values for each variable. These scenarios are TT (true, true), TF (true, false), FT (false, true) and FF (false, false).
03

Create the Truth Table

The truth table will have 3 main sections. The first section will list the truth values of p and q. The second section will list the values of the biconditional \(p \leftrightarrow q\). The third section will contain the values for \(p \leftrightarrow q) \rightarrow q\). The third section values are determined by taking the values of the second section (based on \(p \leftrightarrow q\)) as the input and implication with value of q from first section. Fill out each section according to the rules for each logical connective.
04

Fill in the Truth Table

The completed truth table for the statement \((p \leftrightarrow q) \rightarrow q\) would look like this:| p | q | \(p \leftrightarrow q\) | \((p \leftrightarrow q) \rightarrow q\) ||:---:|:---:|:-----------------:|:-----------------------------:|| T | T | T | T || T | F | F | T || F | T | F | T || F | F | T | F |

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Most popular questions from this chapter

In Exercises 1-24, use Euler diagrams to determine whether each argument is valid or invalid. All writers appreciate language. All poets are writers. Therefore, all poets appreciate language.

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. No journalists learned about the invasion. \(\therefore\) The invasion was a secret.

Use Euler diagrams to determine whether each argument is valid or invalid. All humans are warm-blooded. No reptiles are warm-blooded. Therefore, no reptiles are human.

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 am tired or hungry, I cannot concentrate. I cannot concentrate. \(\therefore\) I am tired or hungry.

Determine whether each argument is valid or invalid. All \(A\) are \(B\), no \(C\) are \(B\), and all \(D\) are \(C\). Thus, no \(A\) are \(D\).
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