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Chapter 3: Problem 19

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

Short Answer

Expert verified
The truth table for the given statement \(\sim(p \leftrightarrow q)\) is as follows: | \(p\) | \(q\) | \(p \leftrightarrow q\) | \(\sim(p \leftrightarrow q)\) ||---|---|---|---|| T | T | T | F || T | F | F | T || F | T | F | T || F | F | T | F |

Step by step solution

01

Understand logical variables and operators

Initially, realize we are dealing with two logical variables, \(p\) and \(q\). The \(\sim\) symbol stands for logical negation (not), and \(\leftrightarrow\) represents logical equivalence (if and only if). The statement \(\sim(p \leftrightarrow q)\) can be interpreted as 'it is not the case that \(p\) if and only if \(q\).'
02

Initialize the truth table

Begin by setting up a truth table with each combination of truth and false values for \(p\) and \(q\). There will be 4 rows (2 ^ number of variables) representing the possible combinations of true (T) and false (F) values.
03

Apply logical equivalence

When \(p\) and \(q\) have the same truth value, i.e., both are true or both are false, the \(p \leftrightarrow q\) is true. Otherwise, it would be false. We add this to the truth table.
04

Apply logical negation

Finally, we apply the logical negation to the result of \(p \leftrightarrow q\). That means, wherever the statement \(p \leftrightarrow q\) is true, now it will be false, and vice versa. This gives us the final truth value of the expression \(\sim(p \leftrightarrow q)\).

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Most popular questions from this chapter

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. \(\sim p \vee q\) P ____ \(\therefore q\)

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 he was disloyal, his dismissal was justified. If he was loyal, his dismissial was justified. \(\therefore\) His dismissal was justified.

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 we close the door, then there is less noise. There is less noise. \(\therefore\) We closed the door.

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

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &q \rightarrow \sim p \\ &q \wedge r \\ &\therefore r \rightarrow p \end{aligned} $$
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