Final answer:
The proposition ( p ∧ q ) → p is the tautology because it is always true, regardless of the truth values of p and q. None of the other options provided satisfy the definition of a tautology.
Step-by-step explanation:
The student's question pertains to the identification of a tautology from a list of logical propositions. By analyzing each proposition, we aim to find one that is always true, regardless of the truth values of its components. A tautology in logic is a compound statement that is true for every possible combination of truth values of its component statements.
Option a ( p ∧ q ) → ¬ p is not a tautology because if both p and q are true, ¬ p would be false, making the implication false.
Option b ( p ∨ q ) → p is not a tautology because if p is false and q is true, the implication would be false.
Option c ( p ∧ q ) ⇔ p is not a tautology because it is equivalent to p → p and q → p; if p is false, the biconditional is false.
Option d ( p ∧ q ) → p is indeed a tautology. This proposition expresses that if both p and q are true, then p is true, which is logically valid regardless of the truth values of p and q. Thus, this implication is always true, satisfying the definition of a tautology.