09B Week 9

# 09B Week 9 - CMPUT 272(Stewart Lecture 16 Reading Epp...

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CMPUT 272 (Stewart) Lecture 16 Reading: Epp Chapter 7 Examples of functions from last time: 1. f : X 7→ Y where X = { a, b, c, d } , Y = { 0 , 1 , 2 , 3 , 4 } , and: f ( a ) = 1 f ( b ) = 2 f ( c ) = 3 f ( d ) = 2 or x f ( x ) a 1 b 2 c 3 d 2 or X a b c d Y 0 1 2 3 4 or f = { ( a, 1) , ( b, 2) , ( c, 3) , ( d, 2) } 2. The function computed by the Division Algorithm is: D : N × Z + 7→ N 2 where each ordered pair ( a, d ) N × Z + maps to the ordered pair ( q, r ) N 2 such that a = dq + r and 0 r < d . 3. mod : Z × Z + 7→ Z where for each integer n and positive integer d , mod ( n, d ) = n mod d = r where n = dq + r, q and r are integers, and 0 r < d. 4. gcd : Z 2 - { (0 , 0) } 7→ Z + where, for all a, b Z that are not both zero, gcd ( a, b ) = the largest integer that divides both a and b. 5. A sequence is a function: 1 1 2 , 1 2 2 , 1 3 2 , 1 4 2 , . . . f : Z + 7→ Q f ( n ) = 1 n 2 6. I X : X 7→ X I X ( x ) = x for all x X The identity function on X 7. C : P ( { r, g, b } ) 7→ N where C ( X ) = | X | 8. B : { 0 , 1 } 4 7→ { 0 , 1 } where B ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 x 2 ) ∧ ∼ ( x 3 x 4 ) 1

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9. For b R + , b 6 = 1, x R + , and y R , the logarithm to the base b of x is the power to which b must be raised to equal x . That is, log b x = y b y = x. log 3 9 = 2 log 2 1024 = 10 log 2 1 2 = - 1 log 99 1 = 0 For all b R + , b 6 = 1, and all x R + : b log b x = x log b b x = x log b 1 = 0 The logarithmic function with base b is the function from R + to R that takes each positive real number x to log b x . Graph of the logarithmic function with base 2: 1 2 4 8 1 2 3 x log 2 x 10. The exponential function with base b is the function from R to R + that takes each real number x to b x . Graph of the exponential function with base 2: 1 2 3 1 2 4 8 x 2 x 2
11. f : R nonneg 7→ R nonneg where f ( x ) = x Every nonnegative real number has a unique nonnegative real square root: its principle square root .

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