ifandonlyif - or the author simply states the extra...

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Structure and Proof of If and Only If Theorems Form of the theorem statement General hypotheses that hold throughout all parts of the theorem Conclusion: A if and only if B (where A and B are statements) This is really a concise way of stating two theorems that are proven separately First theorem Hypotheses: General hypotheses Statement A Conclusion: Statement B This proof sometimes is preceded by the symbol indicating the “direction” of the proof. Second theorem Hypotheses: General hypotheses Statement B Conclusion: Statement A This proof sometimes is preceded by the symbol indicating the “direction” of the proof. Sometimes, instead of preceding one of the directions of the proof with a symbol (
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Unformatted text preview: or ), the author simply states the extra hypothesis (e.g., “Assume Statement A” or “Assume Statement B”). Then, when starting the other direction of the proof, the author might say something like “Conversely assume Statement B” or “Conversely assume Statement A”. The importance of a theorem of this type, once proved, is that Statements A and B are equivalent in the sense that, under the general hypotheses, they either are both true or both false, that is, it is impossible to have one of them be true and the other false. This is a very useful type of result....
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