(8.3)--8.3-ExtensionPrinciplesPart3.pdf

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1、8.38.3 Extension Principles:Part 3 Extension Principles:Part 3 :.:,(),()=,.:,(),()=,.Theorem Theorem 1 1 Let Let:,.Suppose Suppose .Then.Then ()=.=.Theorem Theorem 2 2 Let Let:,.Suppose Suppose,.Then.Then ()().Suppose Suppose,.Then Then ()().Prove that Prove that ()().Proof Proof ,()=(),()().Theorem

2、 Theorem 3 3 Let Let:,.Suppose Suppose is an identifier set,is an identifier set,.ThenThen ()=(),()().Prove that Prove that()=().ProofProof SupposeSuppose .()=()=()()=()().Prove Prove that that()().ProofProof SupposeSuppose .,()(),()()|()().Theorem Theorem 4 4 Let Let:,.Suppose Suppose is an identif

3、ier set,is an identifier set,ThenThen ()=(),()=().Prove that Prove that ()=().ProofProof Suppose Suppose .()=()=()()=()().Theorem Theorem 5 5 Let Let:,.Suppose Suppose .Then.Then ()=().Prove that Prove that()=().ProofProof Suppose Suppose .()=()=()=().Theorem Theorem 6 6 Let Let:,.Let Let ,thenthen

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