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Functions
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L62 Agenda Section 1.8: Functions Domain, co-domain, range Image, pre-image One-to-one, onto, bijective, inverse Functional composition and exponentiation Ceiling “ ” and floor “ ”
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L63 Functions In high-school, functions are often identified with the formulas that define them. EG: f (x ) = x 2 This point of view does not suffice in Discrete Math. In discrete math, functions are not necessarily defined over the real numbers. EG: f (x ) = 1 if x is odd, and 0 if x is even. So in addition to specifying the formula one needs to define the set of elements which are acceptable as inputs, and the set of elements into which the function outputs.
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L64 Functions. Basic-Terms. DEF: A function f : A B is given by a domain set A, a codomain set B, and a rule which for every element a of A, specifies a unique element f (a) in B. f (a) is called the image of a, while a is called the pre-image of f (a). The range (or image) of f is defined by f (A) = {f (a) | a A }.
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L65 Functions. Basic-Terms. EG: Let f : Z R be given by f (x ) = x 2 Q1: What are the domain and co-domain? Q2: What’s the image of -3 ? Q3: What are the pre-images of 3, 4? Q4: What is the range f (Z) ?
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L66 Functions. Basic-Terms. f : Z R is given by f (x ) = x 2 A1: domain is Z, co-domain is R A2: image of -3 = f (-3) = 9 A3: pre-images of 3: none as 3 isn’t an integer! pre-images of 4: -2 and 2 A4: range is the set of perfect squares f (Z) = {0,1,4,9,16,25,…}
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L67 Functions. Sub-ranges. The effect of functions on subsets of the domain is often important. DEF: Given a function f : A B. The pre- image set (or inverse image) of b is defined by f -1 (b) = {a A | f (a)=b }. Given subsets S A and T B, the image set of S is defined by f (S ) = {f(a ) | a S } and the pre-image set (or inverse image) of T is defined by f -1 (T ) = {a A | f (a) T }. NOTE: Even when f is not invertible, the inverse image is defined!
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L68 Functions. Sub-ranges. EG: f : Z R with f (x ) = x 2 Q1: Calculate f –1 (3) Q2: Calculate f –1 (4) Q3: Calculate f ( {-9,-5,-3,0,1,2,3,4} ) Q4: Calculate f –1 ({-9,-5,-3,0,0.25,1,2,2.25,3,4})
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L69 Functions. Sub-ranges. EG: f : Z R with f (x ) = x 2 A1: f –1 (3) = A2: f –1 (4) = {-2, 2} A3: f ( {-9,-5,-3,0,1,2,3,4} ) = {81,25,9,0,1,4,16} A4:f –1 ({-9,-5,-3,0,0.25,1,2,2.25,3,4}) = {0,-1,1,-2,2}
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L610 One-to-One, Onto, Bijection. Intuitively. Represent functions using “node and arrow” notation: One-to-One means that no clashes occur. BAD:a clash occurred, not 1-to-1 GOOD:no clashes, is 1-to-1 Onto means that every possible output is hit BAD: 3 rd output missed, not onto GOOD:everything hit, onto
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L611 One-to-One, Onto, Bijection. Intuitively. Bijection means that when arrows reversed, a function results. Equivalently, that both one-to-one’ness and onto’ness occur. BAD:not 1-to-1. Reverse over-determined: BAD:not onto. Reverse under-determined: GOOD:Bijection. Reverse is a function:
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L612 One-to-One, Onto, Bijection. Formal Definition. DEF: A function f : A B is: one-to-one (or injective) if different elements of A always result in different images in B. onto (or surjective) if every element in B is hit by f. I.e., f (A ) = B. a one-to-one correspondence (or a bijection, or invertible) if f is both one-to-one as well as onto. If f is invertible, its inverse f -1 : B A is well defined by taking the unique element in the pre- image of b, for each b B.
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L613 One-to-One, Onto, Bijection. Examples. Q: Which of the following are 1-to-1, onto, a bijection? If f is invertible, what is its inverse? 1. f : Z R is given by f (x ) = x 2 2. f : Z R is given by f (x ) = 2x 3. f : R R is given by f (x ) = x 3 4. f : Z N is given by f (x ) = |x |
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L614 One-to-One, Onto, Bijection. Examples. 1. f : Z R, f (x ) = x 2 : none 1. not 1-1 clashes for -1,1 in Z 2. not onto -1,-2 missed from R 2. f : Z R, f (x ) = 2x : 1-1 3. f : R R, f (x ) = x 3 : 1-1, onto, bijection, inverse is f (x ) = x (1/3) 4. f : Z N, f (x ) = |x |: onto
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L615 Composition When a function f spits out elements of the same kind that another function g eats, f and g may be composed by letting g immediately eat each output of f. DEF: Suppose that g : A B and f : B C are functions. Then the composite f g : A C is defined by setting f g (a) = f ( g (a) )
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L616 Composition. Examples. Q: Compute g f where 1.f : Z R, f (x ) = x 2 and g : R R, g (x ) = x 3 2. f : Z Z, f (x ) = x + 1 and g = f -1 so g (x ) = x – 1 3. f : {people} {people}, f (x ) = the father of x, and g = f
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L617 Composition. Examples. 1.f : Z R, f (x ) = x 2 and g : R R, g (x ) = x 3 f g : Z R, f g (x ) = x 6 2. f : Z Z, f (x ) = x + 1 and g = f -1 f g (x ) = x (true for any function composed with its inverse) 3. f : {people} {people}, f (x ) = g(x ) = the father of x f g (x ) = grandfather of x from father’s side
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L618 Repeated Composition When the domain and codomain are equal, a function may be self composed. The composition may be repeated as much as desired resulting in functional exponentiation. The whole process is denoted by f n (x ) = f f f f … f (x ) where f appears n –times on the right side. Q1: Given f : Z Z, f (x ) = x 2 find f 4 Q2: Given g : Z Z, g (x ) = x + 1 find g n Q3: Given h(x ) = the father of x, find h n
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L619 Repeated Composition A1: f : Z Z, f (x ) = x 2. f 4 (x ) = x (2*2*2*2) = x 16 A2: g : Z Z, g (x ) = x + 1 g n (x ) = x + n A3: h (x ) = the father of x, h n (x ) = x ’s n’th patrilineal ancestor
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L620 Ceiling and Floor This being a course on discrete math, it is often useful to discretize numbers, sets and functions. For this purpose the ceiling and floor functions come in handy. DEF: Given a real number x : The floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integer greater or equal to x. NOTATION: floor(x) = x , ceiling(x) = x Q: Compute 1.7 , -1.7 , 1.7 , -1.7 .
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L621 Ceiling and Floor A: 1.7 = 1, -1.7 = -2, 1.7 = 2, -1.7 = -1 Prove : show that for all positive real numbers x, y: x.y <= x . y
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