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Chapter 3: Problem 49

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \((p \leftrightarrow q) \leftrightarrow[(q \rightarrow p) \wedge(p \rightarrow q)]\)

Short Answer

Expert verified
By performing these steps on the statement, it will be found that for every row, the final outcomes of \(p \leftrightarrow q\) and \((q \rightarrow p) \wedge (p \rightarrow q)\) match. This indicates that the statement is a tautology.

Step by step solution

01

Identify all the Propositional Variables

The statement given contains two propositional variables \(p\) and \(q\). Since there are two variables, the truth table will have \(2^2 = 4\) rows representing all possible combinations of truth values for \(p\) and \(q\).
02

Construct the Truth Table

Create a truth table with four columns to represent \(p\), \(q\) and the main connectives in the given proposition. Label these columns as: \(p\), \(q\), \(p \leftrightarrow q\) , and \((q \rightarrow p) \wedge (p \rightarrow q)\). Make sure to fill the first two columns with all combinations of truth values for \(p\) and \(q\).
03

Compute the Outcomes for Each Logical Connective

Fill in the truth table by applying the rules of logic to each logical connective. The biconditional operator \(\leftrightarrow\) outputs true when both sentences are the same (either both true or both false). The implication operator \(\rightarrow\) outputs true unless a true proposition implies a false one. The conjunction operator \(\wedge\) outputs true if both of its sentences are true.
04

Compare the Final Outcomes

The last step is to compare the final outcomes of the columns labeled \(p \leftrightarrow q\) and \((q \rightarrow p) \wedge (p \rightarrow q)\). If they are the same in every row, then the whole statement is a tautology. If they are all false, then the statement is a contradiction. If some are true and others are false, then it is neither a tautology nor a contradiction.

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