Consider the following game involving a deck of cards. We are only concerned about the color...
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Consider the following game involving a deck of cards. We are only concerned about the color of the cards, red and black; we ignore the suits and the denominations. The rules for a given game are sequences of "R"s and "B"s corresponding to red and black cards. For example, one game has the three rules: (1) BB, (2) RR, and (3) BRBR. The object of the game is to see if you can use the rules to transform a starting sequence to a goal sequence. At stages along the way, you can apply the rules as follows, as many times as you wish: you can insert the sequence corresponding to the rule anywhere in your current sequence (and you always have enough cards to do this as many times as you would like), or you may remove the sequence corresponding to the rule from your current sequence. For example, we can transform the starting sequence RB to goal sequence BR as follows. From RB we can use Rule (1) to insert BB at the beginning and get BBRB, and then use Rule (2) to insert RR on the end, resulting in BBRBRR. Applying Rule (3) to this last sequence we can remove the four cards in the middle to get BR, so we have gone from RB to BR in three steps using each rule once. a) Suppose the rules are (1) BB, (2) RRR, and (3) RRBRB. Show that you can go from RB to BR, but you cannot go from R to B. b) Suppose the rules are (1) BB, (2) RRR, (3) RBRB. Show that you cannot go from RB to BR. c) For each set of rules in part a) and b), show that every sequence can be transformed into one of the following six, (), B, R, RR, BR, BRR, where () is the "empty sequence" (an empty hand of cards). d) Show that for both the rules in part a), and for the rules in part b), for any sequence x,X2X3 ... Xm there is another sequence y,y2y3 . Yn such that x,x2X3 .. Xmy1Y2Y3 . Yn can be eliminated completely by the rules, resulting in (). Consider the following game involving a deck of cards. We are only concerned about the color of the cards, red and black; we ignore the suits and the denominations. The rules for a given game are sequences of "R"s and "B"s corresponding to red and black cards. For example, one game has the three rules: (1) BB, (2) RR, and (3) BRBR. The object of the game is to see if you can use the rules to transform a starting sequence to a goal sequence. At stages along the way, you can apply the rules as follows, as many times as you wish: you can insert the sequence corresponding to the rule anywhere in your current sequence (and you always have enough cards to do this as many times as you would like), or you may remove the sequence corresponding to the rule from your current sequence. For example, we can transform the starting sequence RB to goal sequence BR as follows. From RB we can use Rule (1) to insert BB at the beginning and get BBRB, and then use Rule (2) to insert RR on the end, resulting in BBRBRR. Applying Rule (3) to this last sequence we can remove the four cards in the middle to get BR, so we have gone from RB to BR in three steps using each rule once. a) Suppose the rules are (1) BB, (2) RRR, and (3) RRBRB. Show that you can go from RB to BR, but you cannot go from R to B. b) Suppose the rules are (1) BB, (2) RRR, (3) RBRB. Show that you cannot go from RB to BR. c) For each set of rules in part a) and b), show that every sequence can be transformed into one of the following six, (), B, R, RR, BR, BRR, where () is the "empty sequence" (an empty hand of cards). d) Show that for both the rules in part a), and for the rules in part b), for any sequence x,X2X3 ... Xm there is another sequence y,y2y3 . Yn such that x,x2X3 .. Xmy1Y2Y3 . Yn can be eliminated completely by the rules, resulting in ().
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Making Hard Decisions with decision tools
ISBN: 978-0538797573
3rd edition
Authors: Robert Clemen, Terence Reilly
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