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Unformatted text preview: Anatomy of a Proof Task: Prove that S ∩ ∅ = ∅ for every set S. 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. 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 ∩ ∅ ....
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This note was uploaded on 02/20/2012 for the course MATH 4000 taught by Professor Staff during the Spring '08 term at UGA.
 Spring '08
 Staff

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