Which statement is always true regardless of p and q?

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Understand the Problem

The question is asking which of the provided logical statements is universally true regardless of the truth values of propositions p and q.

Answer

The correct answer is $p \lor \neg p$.
Answer for screen readers

The universally true statement is: $p \lor \neg p$.

Steps to Solve

  1. Analyze each statement for truth values

We need to evaluate each logical statement to determine if it is universally true regardless of the truth values of propositions $p$ and $q$.

  1. Evaluate statement a: $p \lor \neg p$

This statement represents a logical disjunction (OR) where one of the terms is the negation of $p$. This is a tautology because regardless of whether $p$ is true or false, the entire statement will always be true.

  1. Evaluate statement b: $\neg p \rightarrow p$

This is a conditional statement. If $p$ is true, then the statement is false (since $\neg p$ would be false). If $p$ is false, the statement is true. This is not universally true.

  1. Evaluate statement c: $p \rightarrow q$

This conditional statement states that if $p$ is true, then $q$ must also be true. However, this can be false if $p$ is true and $q$ is false. Thus, it is not universally true.

  1. Evaluate statement d: $p \land \neg p$

This statement represents a logical conjunction (AND) where one of the terms is the negation of $p$. This statement is never true since both $p$ and $\neg p$ cannot be true at the same time.

The universally true statement is: $p \lor \neg p$.

More Information

The statement $p \lor \neg p$ is known as the Law of Excluded Middle in propositional logic. It asserts that for any proposition $p$, either $p$ is true, or its negation $\neg p$ is true, making it a fundamental principle in classical logic.

Tips

Common mistakes include:

  • Confusing the truth values of conditional statements. Remember that an implication $A \rightarrow B$ is only false when $A$ is true, and $B$ is false.
  • Misinterpreting the conjunctions and disjunctions; for example, assuming $p \land \neg p$ can be true.
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