Final answer:
The law that shows the logical equivalence of the two propositions provided is DeMorgan's law. This logical law explains the relationship between the negations of conjunctions and disjunctions. The law of the excluded middle and the law of noncontradiction are fundamental aspects of logical equivalence.
Step-by-step explanation:
The student is asking about logical equivalences and is looking to identify which logical law shows that two propositions are equivalent. The specific proposition given is -((w ∨ p) ∧ (-q ∧ q ∧ w)) = ¬(w ∨ p) ∨ -(-q ∧ q ∧ w). The correct answer to this question is DeMorgan's law, which states that the negation of a conjunction is equivalent to the disjunction of the negations, and the negation of a disjunction is equivalent to the conjunction of the negations. Therefore, the equivalence is shown by transforming the negation of the conjunction into a disjunction of negations.
As part of this explanation, we could mention that a counterexample is a specific case that shows a general statement to be false, which is often used in logic and mathematics to refute universal statements. An example of a conditional might be 'If it rains, the ground gets wet,' where 'it rains' is the sufficient condition, and 'the ground gets wet' is the necessary condition.
The law of the excluded middle states that for any statement, either that statement or its negation is true. This is related to, and logically implied by, the law of noncontradiction, which asserts that a statement and its negation cannot both be true simultaneously.