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  3. roving theorems ma or each of the tollowing statements translate...

Question: roving theorems ma or each of the tollowing statements translate...

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roving theorems ma or each of the tollowing statements, translate it into predicate logic and prove it, if the statement is 1. for any two even integers, there exists a third integer (even or odd) 2. for any two odd integers, there exists a third integer (even or odd) the true, or disprove it, otherwise the double of which is equal to the sum of the first two integers triple of which is equal to the sum of the first two integers Problem 3 (Finding a treasure!) [20 marks] In the back of an old cup- board you discover a note signed by a pirate famous for his bizarre sense of humour and love of logical puzzles. In the note he wrote that he had hid- den treasure somewhere on the property. He listed five true statements and challenged the reader to use them to figure out the location of the treasure 1. If there is an old shipwreck near the beach, then the treasure is buried under a coconut palm tree 2. There is a coconut palm tree growing either at the far end of the island 3. Either there is a shipwreck near the beach, or the treasure is hidden 4. If there is a coconut palm tree at the far end of the island, then there or near the cave n a cave. is no shipwreck on the beach 5. There is no coconut palm tree near the cave Problem 4 (Deciding consistency) [20 marks] A set of propositions is consistent if there is an assignment of truth values to each of the propo- sitional variables, that makes all propositions true. Is the following set of propositions consistent? 1. The system is in multiuser state if and only if it is operating normally 2. If the system is operating normally, the kernel is functioning 3. The kernel is not functioning or the system is in interrupt mode 4. If the system is not in multiuser state, then it is in interrupt mode 5. The system is in interrupt mode Problem 5 (Deciding satisfiability) [20 marks] Let p, q, r be three Boolean variables. For each of the following propositional formulas determine whether it is satisfiable or not

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