(8.2)--8.2-ExtensionPrinciplesPart2.pdf

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1、8.28.2 Extension Principles:Part 2Extension Principles:Part 2 Classical Classical extension extension principleprinciple :.:,:.Theorem Theorem 1 1 Let Let:,.If If .Then Then =.=.Example 1 Example 1 Given Given:,=.Suppose Suppose =|,=.Theorem Theorem 2 2 Let Let:,.Suppose Suppose,.Then Then ()().Supp

2、ose Suppose,.Then.Then ()().Prove thatProve that ()().ProofProof Suppose Suppose .=,=,=.Theorem Theorem 3 3 Let Let:,.Suppose Suppose is an is an identifier identifier set,set,.ThenThen =,.Prove thatProve that .,such that,such that =,such that ,such that ,and,and =,such that,such that ().Prove that

3、Prove that ().,().()=.Prove thatProve that ().,()(),()()|().Theorem Theorem 4 4 Let Let:,.Suppose Suppose is an identifier set,is an identifier set,.ThenThen =,=.Prove that Prove that =.ProofProof (),such that ,such that (),such that,such that ().Theorem 5 Theorem 5 Let Let:,.Suppose Suppose .Then.T

4、hen =().ProofProof ().Theorem 6 Theorem 6 Let Let:,.Let Let ,thenthen ().If If is is injectiveinjective,thenthen =().Proof Proof ().Proof Proof ,=,is injective,is injective,=,.().Prove that if Prove that if is is injectiveinjective,then,then =().Theorem Theorem 7 7 Let Let:,.Let Let ,thenthen ().If If is is surjectivesurjective,thenthen ()=.