# IS11F - Math 300 Fall 2011 Informal Summary(IS Prof Girardi...

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Unformatted text preview: Math 300 - Fall 2011 Informal Summary (IS) Prof. Girardi 0. CHAPTER 0 : Preface to the Student 0.0.1. Remark (Course Goals) . Our main goals in this course are: ◦ to improve our ability to think and reason (useful no matter what career you choice) ◦ to write in a professional mathematical fashion ◦ to gain a solid understand of the material most useful for advanced math courses. ◦ Do you want to add more? Although you might not become a research mathematician, in almost any mathematically related work you may do, the kind of reasoning you need to be able to do is the same reasoning you use in proving theorems. You must first understand exactly what you want to prove (verify, show, or explain), and you must be familiar with the logical steps that allow you to get from the hypothesis to the conclusion. 0.0.2. Remark. Various sets of numbers are excellent sources for developing an understanding of the structure of a correct proof. So the following definitions and remarks, which you may use when forming a proof, will be used extensively in early examples of proof writing. 0.0.3. Definition (Sets and Numbers and their Symbols) . real numbers = R natural numbers = N = { 1 , 2 , 3 , 4 ,... } integers = Z = { , ± 1 , ± 2 , ± 3 , ± 4 ... } rational numbers = Q = n p q ∈ R : p,q ∈ Z and q 6 = 0 o irrational numbers = R \ Q = { x : x ∈ R and x / ∈ Q } complex numbers = C = { a + ib : a,b ∈ R } positive real numbers = R > = { x ∈ R : x > } nonnegative real numbers = R ≥ = { x ∈ R : x ≥ } negative real numbers = R < = { x ∈ R : x < } nonpositive real numbers = R ≤ = { x ∈ R : x ≤ } prime numbers = P { p ∈ N : p 6 = 1 and the only natural numbers that divide p are 1 and p } composite numbers = = { n ∈ N : n 6 = 1 and n / ∈ P } = N \ ( { 1 } ∪ P ) empty set = ∅ = a set with no elements a finite set = = an empty set or a set with n ∈ N elements an infinte set = = a set that is not a finite set 0.0.4. Definition (More on Numbers) . Even/Odd Numbers (1) x ∈ Z is even if and only if there is k ∈ Z such that x = 2 k ⇐⇒ x ∈ 2 Z . (1’) x ∈ N is even if and only if there is k ∈ N such that x = 2 k ⇐⇒ x ∈ 2 N . (2) x ∈ Z is odd if and only if there is j ∈ Z such that x = 2 j- 1 ⇐⇒ x ∈ 2 Z- 1 . (2’) x ∈ Z is odd if and only if there is j ∈ Z such that x = 2 j + 1 ⇐⇒ x ∈ 2 Z + 1 . (2”) x ∈ N is odd if and only if there is j ∈ N such that x = 2 j- 1 ⇐⇒ x ∈ 2 N- 1 . (2”’) x ∈ N is odd and x ≥ 3 if and only if there is j ∈ N such that x = 2 j + 1. Divides a | b ⇐⇒ a divides b ⇐⇒ a is a divisor of b ⇐⇒ b is divisible by a ⇐⇒ b is a multiple of a . (3) For a,b ∈ N : a divides b if and only if there is a k ∈ N such that ak = b ....
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IS11F - Math 300 Fall 2011 Informal Summary(IS Prof Girardi...

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