Final answer:
Using the distributive law of logic, (P and (Q or R)) is shown to be equivalent to ((P and Q) or (P and R)) by distributing the conjunction over the disjunction and vice versa, according to logical identities.
Step-by-step explanation:
The question asked by the student requires proving several logical equivalences without the use of truth tables. To prove that (P ∧ (Q ∨ R)) ≡ ((P ∧ Q) ∨ (P ∧ R)), we can use the distributive law of logic, which states that the conjunction inside the parentheses can be distributed over the disjunction, similar to factoring in algebra. By applying this law in both directions, we can show that these statements are equivalent.
The distributive law like this: (A ∧ (B ∨ C)) ≡ ((A ∧ B) ∨ (A ∧ C)).
We perform the distribution as follows:
- Starting with the left side, (P ∧ (Q ∨ R)), distribute P over (Q ∨ R) to get (P ∧ Q) ∨ (P ∧ R).
- For the right side, ((P ∧ Q) ∨ (P ∧ R)), we can factor out the common P, resulting in P ∧ (Q ∨ R), confirming the equivalence.
The other parts of the question would be approached similarly, checking each expression for logical identities such as commutation, association, distribution, and negation, making the equivalences quite straightforward to prove without truth tables.