Final answer:
To prove the statement by its contrapositive, you must prove choice D: 'If the dog does not get a treat, then the dog will not do a trick.' This method flips and negates both the hypothesis and conclusion of the original statement and is a valid proof technique in logic.
Step-by-step explanation:
To prove the statement “If the dog does a trick, then the dog will get a treat” by proving its contrapositive, you should prove that “If the dog does not get a treat, then the dog will not do a trick.” This is because the contrapositive of a statement flips and negates both the hypothesis and the conclusion. So, the original statement's hypothesis is that the dog does a trick, and the conclusion is that the dog will get a treat. The contrapositive, therefore, would negate both: If the dog does not get a treat (negated conclusion), then the dog did not do a trick (negated hypothesis).
In formal logic, proving the contrapositive is a valid method of proof. The original statement is represented as “If P, then Q” and the contrapositive is “If not Q, then not P.” By proving the contrapositive, you are proving that the absence of the result (Q) implies the absence of the condition (P), which is logically equivalent to the original conditional statement. In this case, choice D is the correct statement to prove.