In the back of an old cupboard you discover a note signed by a pirate famous for his bizarre sense of humor and love of logical puzzles. In the note he wrote that he had hidden treasure somewhere on the property. He lists five statements that are all true and challenges the reader to figure out the location of the treasure from the statements. Here are the statements:

a. If this house is next to a lake, then the treasure is not in the kitchen. b. If the tree in the front yard is an elm, then the treasure is in the kitchen. c. This house is next to a lake. d. The tree in the front yard is an elm or the treasure is buried under the flagpole. e. If the tree in the back yard is an oak, then the treasure is in the garage.

Where is the treasure hidden? Prove your answer using the rules of inference in section 1.6 of the textbook.

TABLE 1 Rules of Inference Rule of InferenceTautology Name (p ^ (p q)) q Modus ponens Modus tollens ((p q) ^ (q r)) (p r) | Hypothetical syllogism Disjunctive syllogisnm Addition (p ^ q) p Simplification Conjunction Resolution ..qvr TABLE 1 Rules of Inference Rule of InferenceTautology Name (p ^ (p q)) q Modus ponens Modus tollens ((p q) ^ (q r)) (p r) | Hypothetical syllogism Disjunctive syllogisnm Addition (p ^ q) p Simplification Conjunction Resolution ..qvr