Copyright © Zeph Grunschlag, 2001-2002. Functions Zeph Grunschlag Copyright © Zeph Grunschlag, 2001-2002.

Agenda Section 1.6: Functions Domain, co-domain, range Image, pre-image One-to-one, onto, bijective, inverse Functional composition and exponentiation Ceiling “ ” and floor “ ” L6

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. L6

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 }. L6

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) ? L6

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,…} L6

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: 3rd output missed, not onto GOOD: everything hit, onto L6

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: L6

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. Alternate definitions using cardinality of pre-image: Injective: |f -1(b)| ≤ 1 for all b  B. Surjective: |f -1(b)|≥ 1 for all b  B. Bijective: |f -1(b)| = 1 for all b  B. L6

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? f : Z  R is given by f (x ) = x 2 f : Z  R is given by f (x ) = 2x f : R  R is given by f (x ) = x 3 f : Z  N is given by f (x ) = |x | L6

One-to-One, Onto, Bijection. Examples. f : Z  R, f (x ) = x 2: none not 1-1 clashes for -1,1 in Z not onto -1,-2 missed from R f : Z  R, f (x ) = 2x : 1-1 f : R  R, f (x ) = x 3: 1-1, onto, bijection, inverse is f (x ) = x (1/3) f : Z  N, f (x ) = |x |: onto L6

Composition stop here http://info.psu.edu.sa/psu/cis/kalmustafa/ 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) ) L6

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 L6

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 L6

f n (x ) = f f f f  … f (x ) 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 hn L6

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 gn (x ) = x + n A3: h (x ) = the father of x, hn (x ) = x ’s n’th patrilineal ancestor L6

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. L6

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  L6