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Chapter 3: Problem 26

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

Short Answer

Expert verified
The final column of the truth table will hold the ultimate truth value of the whole logical statement. This is found by checking the equivalence between the values from the two complex expressions under all scenarios.

Step by step solution

01

Understand the logical operators and notations

Firstly, a deep understanding of the logical operators and notations is necessary. Here are some of them: \n1) \( p \rightarrow q \) is the operator for 'if p, then q.', \n2) \( \sim p \) is the operator for 'not p.', \n3) \( p \vee q \) is the operator for 'p or q.', \n4) \( p \wedge q \) is the operator for 'p and q.', \n5) \( p \leftrightarrow q \) is the operator for 'p if and only if q.'.
02

Construct initial truth table for the variables

Secondly, construct the initial truth table for the variables \( p \) and \( q \). Both \( p \) and \( q \) can be true or false, thus four scenarios are possible. These scenarios are (True,True), (True,False), (False,True), and (False,False).
03

Calculate complex logical expressions

Thirdly, the calculated values of the complex logical expressions \( (p \rightarrow q) \vee(p \wedge \sim p) \) and \( \sim q \rightarrow \sim p \) are added to the truth table. The value of each complex logical expression then depends on the scenarios formulated in the second step.
04

Compare final logical values

Lastly, compare final logical values of the complex logical expressions and add a final column in the truth table for \( [(p \rightarrow q) \vee(p \wedge \sim p)] \leftrightarrow(\sim q \rightarrow \sim p) \). It will be true if and only if both the complex logical expressions return the same truth values. That is, both are either true or false.

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