quantifiers

quantifiers - Quantifiers Professor Sormani A supplement...

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Unformatted text preview: Quantifiers Professor Sormani A supplement for MAT175 Quantifiers are special symbols that are used to make it easier to think about mathematical statements. There are three quantifiers: ∀ means “for all” “for every” ∃ means “there exists” “there is a” ∃ ! means “there exists unique” or “there is only one” The following statements are true. Why? ∀ x ∈ [0, 1], sin(x) is well defined . and ∃ x ∈ [−1, 1] such that 1/x is undefined. and f : [0, 1] → [0, 3] has an inverse if ∀ y ∈ [0, 3] ∃ ! x ∈ [0, 1] such that f (x) = y. Now write true and false next to the following statements after translating them. The answers are below so you can check each answer before continuing. 1) ∃ x ∈ [0, π/2] such that sin(x) = 1. 2) ∃ x ∈ [0, π/4] such that sin(x) = 1. 3) ∃ ! x ∈ [0, π/2] such that sin(x) = 0. 4) ∃ ! x ∈ [0, π ] such that sin(x) = 0. 5) ∀ y ∈ [−1, 1] ∃ x ∈ [0, π/2] such that sin(x) = y. 6) ∀ y ∈ [−1, 1] ∃ x ∈ [0, 2π ] such that sin(x) = y. 7) ∀ y ∈ [−1, 1] ∃ ! x ∈ [0, 2π ] such that sin(x) = y. 8) ∀ y ∈ [−1, 1] ∃ ! x ∈ [−π/2, π/2] such that sin(x) = y. 9) ∀ y ∈ [0, 1] ∃ ! x ∈ [0, π/2] such that sin(x) = y. 10) ∀ y ∈ [0, 1] ∃ x ∈ [0, π/2] such that sin(x) = y. The answers to these questions are: 1) T 4) F 8) T 2) F 5) F 9) T 3) T 6) T 10) T 7) F Try again replacing sin with cos throughout. The answers are different. 1 ...
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This note was uploaded on 01/27/2010 for the course MATH 414 taught by Professor Staff during the Fall '08 term at Indiana.

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