Final Answer:
Using logical equivalences and properties, p→(q→r) is equivalent to q→(p∨r).
Step-by-step explanation:
To demonstrate the equivalence p→(q→r) = q→(p∨r) without a truth table, we'll employ logical equivalences and properties of implication and disjunction.
Starting with p→(q→r), we'll manipulate it using logical equivalences. The implication p→q is logically equivalent to ¬p∨q (the material implication). Similarly, q→r can be represented as ¬q∨r.
Therefore, p→(q→r) becomes p→(¬q∨r). Applying the implication rule again, we have ¬p∨(¬q∨r). Utilizing the associative property of disjunction (p∨q)∨r = p∨(q∨r), we rearrange the expression to (¬p∨¬q)∨r.
The law of commutativity allows us to interchange the order of disjunction, giving (¬q∨¬p)∨r, which is equivalent to q→(¬p∨r). Using the material implication in reverse (q→p = ¬q∨p), we get q→(p∨r), which finally becomes q→(p∨r).
This demonstrates the equivalence p→(q→r) = q→(p∨r) through a series of logical transformations without using a truth table, relying solely on logical equivalences and properties of implication and disjunction.