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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- Spring '11
- Logic, English-language films, Mathematical logic, Mathematical proof