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AnatomyOfAProofSlideshow

AnatomyOfAProofSlideshow - Anatomy of a Proof Task Prove...

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Anatomy of a Proof Task: Prove that S = for every set S.
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Anatomy of a Proof Task: Prove that S = for every set S. Claim: Let S be a set. Then S = . Proof: Let S be a set. ( ) Suppose x S . Then x S and x , so in particular x . Therefore S . ( ) Because has no elements, the statement x is false for all x . Hence the implication “If x , then x S ” is vacuously true. Thus S . Since S and S , we conclude that S = , as desired.
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Task: Prove that S = for every set S. Claim: Let S be a set. Then S = . Proof: Let S be a set. ( ) Suppose x S . Then x S and x , so in particular x . Therefore S . ( ) Because has no elements, the statement x is false for all x . Hence the implication “If x , then x S ” is vacuously true. Thus S . Since S and S , we conclude that S = , as desired.
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