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equivalent - Each arrow corresponds to a single proof in...

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Theorem Proofs for Several Equivalent Conditions In this type of situation the theorem statement is of the following form: Hypotheses Conclusion: Then the following are equivalent: (a) (b) (c) where the number of statements (a), (b), (c), … is usually at least three and can be any number larger than three (if there are only two statements, the theorem is usually presented as an “if and only if” one). What the theorem is stating is that all the conditions are equivalent in the sense that that if any are true, they all are true, and if any are false, they all are false. A proof requires the equivalent of an “if and only if” proof for every pair of the statements. If there are n statements, this means the equivalent of ± n ( n 1) 2 “if and only if” proofs, and, of course, each one requires two directions. Fortunately, it usually doesn’t require this formidable amount of work. The following diagrams indicate just three of the possible approaches when there are five statements:
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Unformatted text preview: Each arrow corresponds to a single proof in which the statement at the tail of the arrow implies the statement at the head of the arrow. Thus in the first diagram, we prove (a) implies (b), then prove (b) implies (c), then (c) implies (d), then (d) implies (e), and finally (e) implies (a). Thus any one of the statements implies all the others by following the string of proofs. Of course, the first proof done in this string can be any one of the five. We can see that the 20 proofs (counting each “if and only if” proof as two proofs because of the two directions) required in the worst case with the five statements (a) through (e) have been reduced to only five proofs. The first “circular” proof technique is common. The other two examples shown require six and seven proofs, not too bad. Convince yourself in these latter two cases that every statement implies every other. A line with two arrowheads indicates a standard “if and only if” proof....
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