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Math135Lecture08StudentNotes

Math135Lecture08StudentNotes - Math 135 Lecture 8 Nested...

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Math 135: Lecture 8: Nested Quantifiers Definition 8.1. Let S and T be two sets. A function f : S T is (or ) if and only if for every y T there exists an x S so that f ( x ) = y . Often S and T are equal to R or are subsets of R . Nested Quantifiers 1. Process quantifiers from left to right. 2. Use existing construction and choose techniques as you proceed from left to right. Example 8.2 (Order of Quantifiers is Important) . x y, y > x “Given any integer x , there exists a larger integer y .” (A statement.) y x, y > x “There exists an integer y which is larger than all integers.” (A statement.) Onto To say that the function f : S T onto means: For every y T there exists x S so that f ( x ) = y . Object: Universe of discourse: Certain property: Something happens: The “something that happens” contains a nested existential quantifier with the following components. Object: Universe of discourse: Certain property: Something happens: Example 8.3. The function g : R R defined by g ( x ) = x 2 is NOT onto. Proof: 1. Let y = - 2 R . Is there an x so that g ( x ) = y ? 2. Since g ( x ) = x 2 there is no x R such that g ( x ) = - 2. 3. Thus g ( x ) is not onto. 1
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Math 135: Lecture 8: Nested Quantifiers Proposition 8.4. The function f : R R defined by f ( x ) = x 3 is onto. Proof of Proposition ??: 1. Let y R . Can we find x so that f ( x ) = y ? 2. Consider x = 3 y . 3. Then f ( x ) = f ( 3 y ) = ( 3 y ) 3 = y as needed. Analysis of Proof The definition of onto uses a nested quantifier. Hypothesis: Conclusion: Core Proof Technique: Nested quantifiers. Preliminaries: Onto as it applies in this situation.
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