To solve this problem, we need to identify which of the given propositions is a tautology, i.e., which one is always true regardless of the truth values of the individual propositions.

Let's go through each option:

A. $q \lor p \lor \neg q$:

• This is a disjunction of $q$, $p$, and $\neg q$.
• Simplification: $q \lor \neg q$ is always true because either $q$ is true or $\neg q$ is true.
• So, this reduces to $\text{True} \lor p$, which is always true regardless of $p$.
• Hence, A is a tautology.

B. $q \lor p \to p \land q$:

• This is a conditional statement, which is equivalent to $\neg(q \lor p) \lor (p \land q)$.
• This does not always hold true, depending on the values of $p$ and $q$. Hence, B is not a tautology.

C. $p \to \neg p$:

• This is a conditional statement that says if $p$ is true, then $\neg p$ must be true.
• This is not always true because if $p$ is true, $\neg p$ is false.
• Hence, C is not a tautology.

D. $p \land (q \lor r)$:

• This is a conjunction, which requires both parts to be true for the entire statement to be true.
• This is not always true because it depends on the values of $p$, $q$, and $r$.
• Hence, D is not a tautology.

Conclusion:

The answer is A. $q \lor p \lor \neg q$, which is a tautology.

Would you like a deeper breakdown of any of the propositions or further details?

Here are 5 related questions to consider:

1. What is the definition of a tautology in logic?
2. How can we prove a proposition is a tautology using truth tables?
3. Can you identify a contradiction in propositional logic?
4. How does the distributive property apply to logical operations?
5. What is the difference between tautology and logical equivalence?

Tip: A tautology in propositional logic is a formula that is true in every possible interpretation, no matter the truth values of the individual components.