Final answer:
To prove that p → q and q → r imply p → r is a tautology, we can use deductive reasoning by assuming p → q and q → r are true and showing that p → r is also true. By using a direct proof, we can demonstrate that whenever p is true, r must also be true, which confirms the tautology.
Step-by-step explanation:
To prove that p → q and q → r imply p → r is a tautology, we can use deductive reasoning. Let's assume that both p → q and q → r are true. If p implies q and q implies r, then it follows that p implies r.
We can prove this using a direct proof. Start with the assumption that p → q and q → r are true. If p → q is true, then whenever p is true, q must also be true. Similarly, if q → r is true, then whenever q is true, r must also be true.
Now, let's assume that p is true. By p → q, we know that q is true. And by q → r, we know that r is true. Therefore, whenever p is true, r must also be true. This confirms that p → q and q → r imply p → r, making it a tautology.