Use equivalence laws to show that $(A ∨ C) ∧ (B ∨ C) ∧ (A ∨ D) ∧ (B ∨ D)$ and $(A ∧ B) ∨ (C ∧ D)$ are logically equivalent.
I am trying to figure out how to go about this problem. The part that confuses me is that there are 4 separate terms. How would you apply the laws on that?
Given $(A ∨ C) ∧ (B ∨ C) ∧ (A ∨ D) ∧ (B ∨ D)$,
we see that using the distributive law on the first two clauses, and on the second two clauses, we simplify the left hand side to:
$$((A \land B) \lor C) \land ((A\land B) \lor D)\tag{1}$$
Using the distributive law again, now on $(1)$, we find that $$((A \land B) \lor C) \land ((A\land B) \lor D)\equiv (A\land B) \lor (C\land D)$$ as desired.