1) [2 marks] Use rules of inference to show that the hypotheses a) Randy works hard b) If Randy works hard then he is a dull boy c) If Randy is a dull boy then he will not get the job implies the conclusion "Randy will not get the job." 2) 12 marks each] For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the p a) "IfI take the day off, it either rains or snows." "I took Tuesday off or I took Thursday off" "It was sunny on Tuesday "It did not snow on Thursday b) "IfI eat spicy foods, then I have strange dreams." "I have strange dreams if there is thunder while I sleep." "I did not have strange dreams." am either clever or lucky." "I am not lucky." "If I am lucky, then I will win the lottery "Every Computer Science major has a personal computer Ralph does not have a personal computer "Ann has a personal computer." c) d) e) What is good for corporations is good for Canada." "What is good for Canada is good for you." "What is good for corporations is for you to buy lots of stuff." t "All rodents gnaw their food." "Mice are rodents. "Rabbits do not gnaw their food." "Bats are not rodents." 3) Prove that proposition If n is any odd integer and is any even integer, then, 2n 3m is even. [Bonus] prove the proposition in (3) by contradiction. Hint: you may appeal to facts such as the sum of two even integer is also even integer; and, the sum of two odd integer is even integer