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Class Notes for September 13, 2010 Do Now: The mapping from the set A to the set B is a function, if for every there is exactly one such that the mapping takes a to b. Review of Homework #1: 1)(a) An algebraic number over the field of rational numbers is the set of complex numbers that are the roots to non-zero polynomial equations that have rational coefficients. *Algebraic numbers over a field F have coefficients in that field. **Algebraic numbers form a field. (b) Is 5-3i an algebraic number? (5-3i)2 - 10(5-3i) + 34= 25 -15i –15i -9 -50 +30i +34= 0 Therefore 5-3i is a root of polynomial equation x2-10x+34=0. 2) (c) In the additive group xm-xn=x(m-n) is analogous to xm/xn=xm-n in the multiplicative group since division is analogous to subtraction and in the additive relationship x is being added (m-n) times and in the multiplicative relationship x is being multiplied (m-n) times. (d) In the multiplicative group = xy ∙ is analogous to (x+y) – (w-z) = (x+y) + (z-w) in the additive group since + and – are inverse operations in the additive group and / and ∙ are inverse operations in the multiplicative group. Also in the additive group (w-z) and (z-w) are additive inverses and in the multiplicative group and are multiplicative inverses. (e) In the multiplicative group (bc)a =baca is analogous to a(b+c)=ab+ac in the additive group since + is analogous to ∙ and in the additive relationship (b+c) is being added a times and in the multiplicative relationship (bc) is being multiplied a times. *Similarly ab+c = abac is also analogous to the additive relationship since a is being multiplied (b+c) times. (f) In the additive group 0∙b =0 is analogous to b0=1 in the multiplicative group since in the additive relationship b is being added 0 times to get the additive identity and in the multiplicative group b is being multiplied 0 times to get the multiplicative identity. Key Points When Writing Mathematics: Expression: vs. Equation: 5x-9 5x-9=3 Roots vs. Factors x2-1 (x+1)(x-1) [factors] x=1 and x=-1 [roots] ═ >: “implies” or “If …then…” NOT “then” **DO NOT write ‘if A ═ > B’ for ‘if A then B’ Equivalence Relations and Equivalence Classes An equivalence relation, ~, on a set, S, is a relation that satisfies: 1) Reflexivity: x~x, 2) Symmetry: x~y ═ > y~x 3) Transitivity: x~y and y~z ═ > x~z An equivalence class is a partition on a set, S, such that are in the same equivalence class if x~y. Any two equivalence classes are either equal or disjoint. Example 1: α~β iff sinα=sinβ and cosα=cosβ is an equivalence class on the set of all angles since: 1) Reflexivity: sinα=sinα and cosα=cosα ═ > α~α 2) Symmetry: α~β ═ > sinα=sinβ and cosα=cosβ. By symmetry of =, sinβ=sinα and cosβ=cosα ═ > β~α 3) Transitivity: α~β and β~θ ═ > sinα=sinβ, cosα=cosβ, sinβ=sinθ and cosβ=cosθ ═ >by transitivity of =, sinα=sinθ and cosα=cosθ ═ > α~θ There are infinitely many equivalence classes for this equivalence relation. They are represented by 0 ≤ α < 360. Example 2: On the set of integers, a≡b mod 3 if a-b=3∙q, is an equivalence relation since: 1) Reflexivity: a-a=0=3∙0. Since , a≡ a mod 3. 2) Symmetry: a≡b mod 3 ═ > a-b=3∙q. –(a-b)=-(3∙q) ═> b-a=3∙(-q). Since , b≡a mod 3. 3) Transitivity: a≡b mod 3 and b≡c mod 3 ═ > a-b=3∙q and b-c=3∙z; . c=b-3∙z ═ > a-c=a-(b-3∙z)=a-b+3∙z=3∙q+3∙z=3(q+z). But so a≡c mod 3. The set of equivalence classes mod 3 are: {…,-3n,…,-6, -3, 0, 3, 6,…,3n,…} {…,-3n+1,…,-5, -2, 1, 4, 7,…,3n+1,…} {…,-3n+2,…,-4, -1, 2, 5, 8,…,3n+2,…} or we write Z3={0,1,2} for the set of equivalence classes. Example 3: On the set of integers, a~b if a-b is even is an equivalence relation proved in Homework#1. The equivalence classes for this relation is the set of even integers and the set of odd integers or Z2={0,1}. Extra Example: On the set of all functions f: R->R, f~g if f is a constant non-zero multiple of g, f(x)= k∙g(x) is an equivalence relation since: 1) Reflexivity: f(x)=1∙f(x) ═ > f~f 2) Symmetry: f~g ═> f(x)= k∙g(x) ═> g(x) = ∙ f(x) ═ > g~f 3) Transitivity: f~g and g~t ═ > f(x)= k∙g(x) and g(x)= c∙t(x) ═ > f(x)= k∙(c∙t(x)) ═ > f(x) = (k∙c)∙t(x) ═ > f~t How many equivalence classes are there for this relationship? Reference Angles A reference angle of an angle θ is an angle α with 0 ≤ α ≤ 90 such that = and |cosθ| = |cosα|. The reference angle is always the smallest angle between the terminal side of an angle and the x-axis. A reference angle is a representative of an equivalence class formed by the equivalence relation on the set of all angles: α~θ if|sinθ| = |sinα| and |cosθ| = |cosα| on the set of all angles This relationship creates infinitely many equivalence classes represented by 0 ≤ α < 90. or example a 30 reference angle is a representa ve for its class {30 +k , 150 +k : }.

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posted: | 9/14/2012 |

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