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

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MA1100 Lecture 17 Functions Definitions & Notations Representation of functions xamples of functions Examples of functions Sets of functions Lecture 17 Chartrand: 9.1, 9.2
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t Announcement est scripts - turned during tutorial ± Test scripts returned during tutorial classes this week ± Test solutions vailable in workbin est so ut o s aaa b e o b ± Test Score – will be available in Gradebook later today Lecture 17 2
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i 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 or all a b if 2 b 2 mod 3 For all a, b œ Z , if a + b ª 0 mod 3 , then a ª 0 mod 3 and b ª 0 mod 3 . ª mod n and only if ] = ] Rephrase as congruence class or all [a] [b] if ] 2 ] 2 = [0] a b mod n if and only if [a] n [b] n Lecture 17 3 For all [a] 3 , [b] 3 œ Z 3 , if [a] 3 + [b] 3 [0] 3 , then [a] 3 =[0] 3 and [b] 3 3 .
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i i ibilit Pr bl From lecture 16 Divisibility Problem For all [a] , [b] œ Z , if [a] 2 + [b] 2 = [0] , Proof ] ] ] ÿ ] ] ] 3 3 3 3 3 3 then [a] 3 =[0] 3 and [b] 3 3 . + [0] [1] [2] [0] [0] [1] [2] ] ] ] ] [0] [1] [2] [0] [0] [0] [0] ] ] ] ] [1] [1] [2] [0] [2] [2] [0] [1] [1] [0] [1] [2] [2] [0] [2] [1] [n] 3 2 = diagonal = [0] 3 or [1] 3 [a] 3 2 +[b] 3 2 = [0] 3 cases [a] 3 2 = [0] 3 [b] 3 2 = [0] 3 Lecture 17 4 3 cases [a] 3 2 = [1] 3 [b] 3 2 = [2] 3 [a] 3 2 = [2] 3 [b] 3 2 = [1] 3 impossible [a] 3 = [0] and [b] 3 = [0]
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t i Functions hich of the following are functions? Which of the following are functions? 1. f(x) = sin x 2. f(x) = x 2 3. f(x) = 1/x x 0 4. f(x) = absolute value of x 5. f(x) = square root of x x ¥ 0 ere are two possible square roots x ) x ( f = function has xactly ne output for each input there are two possible square roots Lecture 17 5 A function has exactly one output for each input.
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A function is completely defined when the rule, domain and codomain are given. t i 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 . le The set A is called the domain of the function. The set B is called the codomain of the function. set A set B rule Lecture 17 6 x y Domain Codomain
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t i 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) the domain of this relation is the whole of set A itself . {a b c} {1 2 3} xample R 1 = {(a, 1), (b, 1), (c, 3)} = {(a, 1), (a, 2), (b, 1), (c, 3)} A = {a, b, c} B = {1, 2, 3} Example Lecture 17 7 R 2 {(a, 1), (a, 2), (b, 1), (c, 3)} R 3 = {(a, 1), (a, 2), (c, 3)}
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t i every element of the domain associates to exactly one element of the codomain Functions Example sin x real functions domain codomain rule p sin x R R 0 p /2 0 1 omain odomain rule 1/x R R R –{0} Example 1/x domain codomain / 2 1/2
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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

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