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Chapter 3: Problem 44

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

Short Answer

Expert verified
As per the results of the truth table which maps all values of given statements, we can affirm that \(\sim(p \vee q) \leftrightarrow(\sim p \wedge \sim q)\) is a tautology because each side of the \(\leftrightarrow\) matches for every possible combination of p and q.

Step by step solution

01

Understand the Symbols

The symbols stand for: \(\sim\) - NOT, \(\vee\) - OR, \(\wedge\) - AND, and \(\leftrightarrow\) - IF AND ONLY IF or EQUIVALENCE.
02

Create the Truth Table

It is needed to create a truth table in order to visualize all possible combinations of truth values for p and q. To do so, you should list all combinations of True (T) and False (F) for the variables p and q. Then, you'll carry out the operations \(\sim(p \vee q)\) and \(\sim p \wedge \sim q\) for each combination and write the result in the table.
03

Analyze the Truth Table

After obtaining a truth table, we need to analyze whether the statements on each side of the \(\leftrightarrow\) have the same values for all combinations of p and q, or not. If they do, the whole statement is a tautology. If all values are false, the statement is a self-contradiction. If some are true and others false, the statement is neither.

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