section_2.14_answers

section_2.14_answers - PHIL12A Section answers 14 February...

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PHIL12A Section answers, 14 February 2011 Julian Jonker 1 How much do you know? 1. You should understand why a truth table is constructed the way it is: why are the truth values listed in the order they are? In principle, it doesn’t matter how they are listed, but there is a natural way to list them systematically, which we use. (a) My favourite slot machine gives me money if I line up four cherries in a row. When I pull the lever, each of the four displayed items can come up as either a cherry or a banana. How many different combinations can the machine display? List them. This is exactly like listing all possible truth values for four atomic sentences, except with say, cherry for True and banana for False . There are 2 × 2 × 2 × 2 = 2 4 = 16 possible combinations. Think of it in the following way: Suppose you were betting on just one item. It could come up as a cherry or banana, so there are two possible ‘combinations’. Now suppose you were betting on two items. The first item can be a cherry, but there are two possible ways for it to be a cherry if we include the possible values of the second item: cherry cherry cherry banana But this is true also if it comes up as a banana. banana cherry banana banana So there are four possible combinations. More particularly, we have 2 × 2 = 2 2 = 4 possibilities. If we add a third item, then the first item can come up in four different ways: if it comes up as a cherry, then there are two different ways the second item can come up (cherry or banana), and for each way the second item comes up there are two different ways the third item can come up. So all in all there are 2 × 2 × 2 = 2 3 = 8 possibilities. The same holds for a truth table. If you have n atomic sentences, you will have 2 n rows in your truth table. In other words, the number of rows grows exponentially, which is why you will never be asked to do a truth table for more than a handful of atomic sentences! 1
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(b) Suppose the machine displays three different items at a time – my goal is to line up three cherries. But each item could be either a cherry, a banana, a palm tree or a robot. How many combinations are possible? It would take a long time to list them all, but convince yourself that there is a way to list them systematically so that you are sure you don’t leave any out. What would the 17th item on your list be? There are 4 × 4 × 4 = 4 3 = 64 possible combinations. Think of the list as a sort of tree. For the first item, draw four branches for the different possibilities: cherry, banana, palm tree, robot. At the end of the cherry branch, draw another four branches for the possible values of the second item (cherry, banana, palm tree, robot), and then do this three more times at the end of each of the other branches. You should now have sixteen end points. At the end of each end point, draw four branches for the possible values of the third item. All in all you should have 64 end points now, which you could number if you wish. If you drew
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