Beck Gibson
06/06/2024 · Elementary School
Which of the following is a tautology? \( \begin{array}{l}p \wedge \neg p \\ p \vee \neg p \\ p \Rightarrow \neg p \\ \neg(p \Leftrightarrow p)\end{array} \)
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Step-by-step Solution
To determine which of the given statements is a tautology, we need to analyze each one.
1. **\( p \wedge \neg p \)**: This statement asserts that \( p \) is true and \( p \) is false at the same time. This is a contradiction and is never true. Therefore, it is not a tautology.
2. **\( p \vee \neg p \)**: This statement asserts that either \( p \) is true or \( p \) is false. This is always true regardless of the truth value of \( p \). Therefore, this is a tautology.
3. **\( p \Rightarrow \neg p \)**: This statement asserts that if \( p \) is true, then \( p \) is false. This is only true when \( p \) is false (since if \( p \) is true, \( \neg p \) is false, making the implication false). Therefore, this is not a tautology.
4. **\( \neg(p \Leftrightarrow p) \)**: The biconditional \( p \Leftrightarrow p \) is always true (since \( p \) is equivalent to itself). Therefore, \( \neg(p \Leftrightarrow p) \) is always false. This is not a tautology.
From the analysis, the only tautology among the options is:
\[
\boxed{p \vee \neg p}
\]
Quick Answer
The tautology is \( p \vee \neg p \).
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