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Chapter 3: Problem 31

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

Short Answer

Expert verified
The completed truth table will list all 8 possible combinations of truth values for \(p\), \(q\), and \(r\), and based on these, it will provide the correct truth values for the smaller statements \(\sim p\), \(\sim q\), \(\sim q \wedge p\), \(r \vee(\sim q \wedge p)\), and the full statement \([r \vee(\sim q \wedge p)] \leftrightarrow \sim p\).

Step by step solution

01

Initialize the table

Firstly, a truth table needs to be set up. This table should include columns for the variables \(p\), \(q\), and \(r\), the negation \(\sim p\), and \(\sim q\), the conjunction \(\sim q \wedge p\), and the disjunction \(r \vee(\sim q \wedge p)\), and finally the biconditional \([r \vee(\sim q \wedge p)] \leftrightarrow \sim p\).
02

Fill out the truth values for \(p\), \(q\), and \(r\)

Begin by writing out the truth values for the variables \(p\), \(q\), and \(r\) in their columns. For three variables, there are \(2^3 = 8\) possible combinations of truth values. Half of them will make each variable true, and the other half will make each variable false.
03

Fill out the truth values for \(\sim p\) and \(\sim q\)

Next, one can find the truth values of the negations \(\sim p\) and \(\sim q\). This can be done by simply reversing the truth values of \(p\) and \(q\). If \(p\) or \(q\) is true, their negation is false, and vice versa.
04

Fill out the truth values for \(\sim q \wedge p\)

Then, calculate the truth values of the conjunction \(\sim q \wedge p\). A conjunction is true only if both its components are true. So, the truth value of \(\sim q \wedge p\) is true only when both \(\sim q\) and \(p\) are true.
05

Fill out the truth values for \(r \vee(\sim q \wedge p)\)

Next, calculate the truth values of the disjunction \(r \vee(\sim q \wedge p)\). A disjunction is true if either or both of its components are true. So, in the cases where \(r\) is true, or \(\sim q \wedge p\) is true, or both are true, \(r \vee(\sim q \wedge p)\) will be true.
06

Fill out the truth values for \([r \vee(\sim q \wedge p)] \leftrightarrow \sim p\)

Lastly, find the truth values of the biconditional \([r \vee(\sim q \wedge p)] \leftrightarrow \sim p\). A biconditional is true if both its components have the same truth value. Thus, where \(r \vee(\sim q \wedge p)\) and \(\sim p\) are both true or both false, \([r \vee(\sim q \wedge p)] \leftrightarrow \sim p\) will be true. In all other cases, it will be false.

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Most popular questions from this chapter

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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