Which statement is always true regardless of p and q?

Steps to Solve

1. Identify the Statements List the logical statements provided in the question:

   • a. $p \lor \neg p$
   • b. $\neg p \to p$
   • c. $p \to q$
   • d. $p \land \neg p$

2. Evaluate Each Statement Determine the truth value of each statement regardless of the truth values of $p$ and $q$.

3. Analyze Statement a: $p \lor \neg p$ This is an example of the Law of Excluded Middle. It states that either $p$ is true or $\neg p$ (not $p$) is true.

   • Truth Table:

     $p$	$\neg p$	$p \lor \neg p$
     T	F	T
     F	T	T

   • Conclusion: Always true.

4. Analyze Statement b: $\neg p \to p$ This represents a conditional statement. It's true unless $\neg p$ is true and $p$ is false.

   • Truth Table:

     $p$	$\neg p$	$\neg p \to p$
     T	F	T
     F	T	F

   • Conclusion: Not always true.

5. Analyze Statement c: $p \to q$ This is also a conditional statement. It can be false if $p$ is true and $q$ is false.

   • Truth Table:

     $p$	$q$	$p \to q$
     T	T	T
     T	F	F
     F	T	T
     F	F	T

   • Conclusion: Not always true.

6. Analyze Statement d: $p \land \neg p$ This is a contradiction because $p$ cannot be both true and false at the same time.

   • Truth Table:

     $p$	$\neg p$	$p \land \neg p$
     T	F	F
     F	T	F

   • Conclusion: Always false.

7. Final Conclusion The statement that is always true regardless of the truth values of $p$ and $q$ is option a: $p \lor \neg p$.