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lecture17(complete)

# lecture17(complete) - MA1100 Lecture 17 Functions...

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MA1100 Lecture 17 Functions Definitions & Notations Representation of functions Examples of functions Sets of functions Chartrand: 9.1, 9.2

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A t Announcement Test scripts - returned during tutorial Test scripts classes this week Test solutions available in workbin est so ut o s a a ab e o b Test Score – will be available in Gradebook later today Lecture 17 2
Di i ibilit Pr bl From lecture 16 Divisibility Problem Problem For all a, b œ Z , if 3 divides a 2 + b 2 , then 3 divides a and 3 divides b . Rephrase as congruence modulo For all a b œ Z if a 2 + b 2 ª 0 mod 3 For all a, b , if 0 mod 3 , then a ª 0 mod 3 and b ª 0 mod 3 . a ª b mod n if and only if [a] n = [b] n Rephrase as congruence class For all [a] 3 , [b] 3 œ Z 3 , if [a] 3 2 + [b] 3 2 = [0] 3 , [b] Lecture 17 3 [0] then [a] 3 = [0] 3 and [b] 3 = [0] 3 .

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Di i ibilit Pr bl From lecture 16 Divisibility Problem For all [a] 3 , [b] 3 œ Z 3 , if [a] 3 2 + [b] 3 2 = [0] 3 , Proof + [0] [1] [2] ÿ [0] [1] [2] then [a] 3 = [0] 3 and [b] 3 = [0] 3 . [0] [0] [1] [2] [1] [1] [2] [0] [0] [0] [0] [0] [1] [0] [1] [2] [2] [2] [0] [1] [2] [0] [2] [1] 2 [n] 3 = diagonal = [0] 3 or [1] 3 [a] 3 2 + [b] 3 2 = [0] 3 3 cases [a] 3 2 = [0] 3 [b] 3 2 = [0] 3 2 2 [a] [0] and [b] [0] Lecture 17 4 [a] 3 = [1] 3 [b] 3 = [2] 3 [a] 3 2 = [2] 3 [b] 3 2 = [1] 3 impossible 3 = [0] and [b] 3 = [0]
F ti Functions Which of the following are functions? 1. f(x) = sin x 2. f(x) = x 2 3. f(x) = 1/x 4 f( ) b l l f x 0 4. f(x) = absolute value of x 5. f(x) = square root of x x ¥ 0 there are two possible square roots x ) x ( f = A function has exactly one output for each input Lecture 17 5 one output for each input.

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A function is completely defined when the rule, domain and codomain are given. F ti Functions Definition A function from a set A to a set B is a rule that associate with every element x of A exactly one element of the set B. A function is also called a mapping . rule The set A is called the domain of the function. The set B is called the codomain of the function. set A set B x y Lecture 17 6 Domain Codomain
F ti Functions A function from a set A to a set B is a special type of relation from A to B. In this relation, (1) every element x of A is associated with exactly one element of the set B; (2) th d i f thi l ti i th h l f t A (2) the domain of this relation is the whole of set A itself . A {a b c} B {1 2 3} Example R 1 = {(a, 1), (b, 1), (c, 3)} R 2 = {(a, 1), (a, 2), (b, 1), (c, 3)} A = {a, b, c} B = {1, 2, 3} Lecture 17 7 {(a, 1), (a, 2), (b, 1), (c, 3)} R 3 = {(a, 1), (a, 2), (c, 3)}

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F ti every element of the domain associates to exactly one element of the codomain Functions Example sin x real functions domain codomain rule sin x R R 0 0 p /2 1 domain codomain rule 1/x R R – {0} Example 1/x 0 2 ? 1/2 i l ti Lecture 17 8 some element of set A is not associated to element in set B violation
F ti every element of the domain associates to exactly one element of the codomain Functions Example square root of x domain codomain rule square root of x R + 2 i l ti R + 2 - 2 some element of A is associated to more than

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lecture17(complete) - MA1100 Lecture 17 Functions...

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