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MAT 3749 Introduction to Analysis Section 1.3 Part I Countable Sets http://myhome.spu.edu/lauw
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Goals Review and Renew the concept of functions How to show that a function is an One-to- one function (Injection) How to show that a function is an Onto function (Surjection) Countable and Uncountable Sets
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References Section 1.3 Howland, Appendices A-C
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You Know a Lot About Functions You are supposed to know a lot… Domain, Range, Codomain Inverse Functions One-to-one, Onto Functions Composite Functions
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Is this a Function? (I)
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Is this a Function? (II)
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One-to-One Functions
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Equivalent Criteria
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Example 1 Determine if the given function is injective. Prove your answer.
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Onto Functions
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Equivalent Criteria
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Example 2 Determine if the given function is surjective. Prove your answer.
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Counting Problems…
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Bijections
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Inverse Functions
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Equivalent Sets
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Example 3 The set of odd integers (O) and even integers (E) are equivalent. Plan: 1. Define a function from O to E. 2. Show that the function is well defined. 3. Show that the function is bijective.
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Countable Sets
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Remark
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Theorem
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Analysis
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Proof
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Corollary (HW)
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Theorem
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Proof Outline
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