Vitamin E is an antioxidant and a useless food supplement if and only if it does not reduce heart disease. It is not the case either that vitamin E does not reduce heart disease or is not an antioxidant. Therefore, vitamin E is not a useless food supplement. (A, U, R)

Translate the following arguments into symbolic form and then use the eighteen rules of inference to derive the conclusion of each. Use the translation letters in the order in which they are listed.