Discrete Math for Computer Science CSC 281 Dr.Yuan Tian ytian@ksu.edu.sa Module Topic Basic Structures: Functions

Acknowledgement Most of these slides were either created by Professor Bart Selman at Cornell University or else are modifications of his slides

Functions f(x) x f(x) = -(1/2)x – 1/2 Suppose we have: How do you describe the yellow function? f(x) = -(1/2)x – 1/2 What’s a function ?

Functions Kathy Michael Toby John Chris Brad Carol Mary B A A = {Michael, Toby , John , Chris , Brad } B = { Kathy, Carla, Mary} Let f: A  B be defined as f(a) = mother(a). Michael Toby John Chris Brad Kathy Carol Mary A B

Functions A function f from a set A to a set B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f: AB (note: Here, ““ has nothing to do with if… then) 9/16/2018

Functions Note: Functions are also called mappings or transformations. More generally: B Definition: Given A and B, nonempty sets, a function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the element of B assigned by function f to the element a of A. If f is a function from A to B, we write f : AB. Note: Functions are also called mappings or transformations.

Functions A - Domain of f B- Co-Domain of f domain More generally: f: RR, f(x) = -(1/2)x – 1/2 domain co-domain

Functions If f:AB, we say that A is the domain of f and B is the codomain of f. If f(a) = b, we say that b is the image of a and a is the pre-image of b. The range of f:AB is the set of all images of elements of A. We say that f:AB maps A to B. 9/16/2018

Functions a collection of points! a point! Why not? More formally: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f. a point! A B B A Why not?

Functions Let us take a look at the function f:PC with P = {Linda, Max, Kathy, Peter} C = {Boston, New York, Hong Kong, Moscow} f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = New York Here, the range of f is C. 9/16/2018

Functions Let us re-specify f as follows: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston Is f still a function? yes What is its range? {Moscow, Boston, Hong Kong} 9/16/2018

Functions If the domain of our function f is large, it is convenient to specify f with a formula, e.g.: f:RR f(x) = 2x This leads to: f(1) = 2 f(3) = 6 f(-3) = -6 … 9/16/2018

Functions Other ways to represent f: Boston Peter Hong Kong Kathy Max Moscow Linda f(x) x Linda Max Kathy Peter Boston New York Hong Kong Moscow 9/16/2018

Functions Let f1 and f2 be functions from A to R. Then the sum and the product of f1 and f2 are also functions from A to R defined by: (f1 + f2)(x) = f1(x) + f2(x) (f1f2)(x) = f1(x) f2(x) Example: f1(x) = 3x, f2(x) = x + 5 (f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5 (f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x 9/16/2018

Functions We already know that the range of a function f:AB is the set of all images of elements aA. If we only regard a subset SA, the set of all images of elements sS is called the image of S. We denote the image of S by f(S): f(S) = {f(s) | sS} 9/16/2018

Functions Let us look at the following well-known function: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston What is the image of S = {Linda, Max} ? f(S) = {Moscow, Boston} What is the image of S = {Max, Peter} ? f(S) = {Boston} 9/16/2018

Functions - image & preimage image(S) For any set S  A, image(S) = {b : a  S, f(a) = b} So, image({Michael, Toby}) = {Kathy} image(A) = B - {Carol} range of f image(A) Michael Toby John Chris Brad Kathy Carol Mary B A image(John) = {Kathy} pre-image(Kathy) = {John, Toby, Michael}

Properties of Functions A function f:AB is said to be one-to-one (or injective), if and only if x, yA (f(x) = f(y)  x = y) In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B. 9/16/2018

Properties of Functions g(Linda) = Moscow g(Max) = Boston g(Kathy) = Hong Kong g(Peter) = New York Is g one-to-one? Yes, each element is assigned a unique element of the image. And again… f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston Is f one-to-one? No, Max and Peter are mapped onto the same element of the image. 9/16/2018

Every b  B has at most 1 preimage. Functions - injection A function f: A  B is one-to-one (injective, an injection) if a,b,c, (f(a) = b  f(c) = b)  a = c Not one-to-one Michael Toby John Chris Brad Kathy Carol Mary

Properties of Functions How can we prove that a function f is one-to-one? Whenever you want to prove something, first take a look at the relevant definition(s): x, yA (f(x) = f(y)  x = y) Example: f:RR f(x) = x2 Disproof by counterexample: f(3) = f(-3), but 3  -3, so f is not one-to-one. 9/16/2018

Properties of Functions … and yet another example: f:RR f(x) = 3x One-to-one: x, yA (f(x) = f(y)  x = y) To show: f(x)  f(y) whenever x  y x  y 3x  3y f(x)  f(y), so if x  y, then f(x)  f(y), that is, f is one-to-one. 9/16/2018

Properties of Functions A function f:AB with A,B  R is called strictly increasing, if x,yA (x < y  f(x) < f(y)), and strictly decreasing, if x,yA (x < y  f(x) > f(y)). Obviously, a function that is either strictly increasing or strictly decreasing is one-to-one. 9/16/2018

Properties of Functions A function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b. In other words, f is onto if and only if its range is its entire codomain. A function f: AB is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto. Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|. 9/16/2018

Functions - surjection Every b  B has at least 1 preimage. Functions - surjection A function f: A  B is onto (surjective, a surjection) if b  B, a  A f(a) = b Not onto Michael Toby John Chris Brad Kathy Carol Mary

Functions – one-to-one-correspondence or bijection A function f: A  B is bijective if it is one-to-one and onto. Every b  B has exactly 1 preimage. Anna Mark John Paul Sarah Carol Jo Martha Dawn Eve Anna Mark John Paul Sarah Carol Jo Martha Dawn Eve An important implication of this characteristic: The preimage (f-1) is a function! They are invertible.

Properties of Functions Examples: In the following examples, we use the arrow representation to illustrate functions f:AB. In each example, the complete sets A and B are shown. 9/16/2018

Properties of Functions Is f injective? No. Is f surjective? Is f bijective? Linda Max Kathy Peter Boston New York Hong Kong Moscow 9/16/2018

Properties of Functions Linda Max Kathy Peter Boston New York Hong Kong Moscow Is f injective? No. Is f surjective? Yes. Is f bijective? Paul 9/16/2018

Properties of Functions Linda Max Kathy Peter Boston New York Hong Kong Moscow Lübeck Is f injective? Yes. Is f surjective? No. Is f bijective? 9/16/2018

Properties of Functions Linda Max Kathy Peter Boston New York Hong Kong Moscow Lübeck Is f injective? No! f is not even a function! 9/16/2018

Properties of Functions Linda Boston Is f injective? Yes. Is f surjective? Is f bijective? Max New York Kathy Hong Kong Peter Moscow Helena Lübeck 9/16/2018

Functions: inverse function Definition: Given f, a one-to-one correspondence from set A to set B, the inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b. The inverse function is denoted f-1 . f-1 (b)=a, when f(a)=b. B

Inversion An interesting property of bijections is that they have an inverse function. The inverse function of the bijection f:AB is the function f-1:BA with f-1(b) = a whenever f(a) = b. 9/16/2018

Inversion Example: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Lübeck f(Helena) = New York Clearly, f is bijective. The inverse function f-1 is given by: f-1(Moscow) = Linda f-1(Boston) = Max f-1(Hong Kong) = Kathy f-1(Lübeck) = Peter f-1(New York) = Helena Inversion is only possible for bijections (= invertible functions) 9/16/2018

Inversion Linda Boston f Max New York f-1 f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York. Kathy Hong Kong Peter Moscow Helena Lübeck 9/16/2018

Functions - examples yes yes yes This function is invertible. Suppose f: R+  R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective? yes yes yes This function is invertible.

Functions - examples no yes no This function is not invertible. Suppose f: R  R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective? no yes no This function is not invertible.

Functions - examples no no no Suppose f: R  R, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective? no no no

Functions - composition “f composed with g” Let f: AB, and g: BC be functions. Then the composition of f and g is: (f o g)(x) = f(g(x)) Note: (f o g) cannot be defined unless the range of g is a subset of the domain of f.

Composition The composition of two functions g:AB and f:BC, denoted by fg, is defined by (fg)(a) = f(g(a)) This means that first, function g is applied to element aA, mapping it onto an element of B, then, function f is applied to this element of B, mapping it onto an element of C. Therefore, the composite function maps from A to C. 9/16/2018

Composition Example: f(x) = 7x – 4, g(x) = 3x, f:RR, g:RR (fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101 (fg)(x) = f(g(x)) = f(3x) = 21x - 4 9/16/2018

Composition Composition of a function and its inverse: (f-1f)(x) = f-1(f(x)) = x The composition of a function and its inverse is the identity function i(x) = x. 9/16/2018

Composition Example: Let f(x) = 2 x +3; g(x) = 3 x + 2; (f o g) (x) = f(3x + 2) = 2 (3 x + 2 ) + 3 = 6 x + 7. (g o f ) (x) = g (2 x + 3) = 3 (2 x + 3) + 2 = 6 x + 11. As this example shows, (f o g) and (g o f) are not necessarily equal – i.e, the composition of functions is not commutative.

Composition Note: (f -1 o f) (a) = f -1(f(a)) = f -1(b) = a. (f o f -1) (b) = f (f -1(b)) = f-(a) = b. Therefore (f-1o f ) = IA and (f o f-1) = IB where IA and IB are the identity function on the sets A and B. (f -1) -1= f

Some important functions Absolute value: Domain R; Co-Domain = {0}  R+ |x| = x if x ≥0 -x if x < 0 Ex: |-3| = 3; |3| = 3 Floor function (or greatest integer function): Domain = R; Co-Domain = Z x  = largest integer not greater than x Ex: 3.2 = 3; -2.5 =-3

Some important functions Ceiling function: Domain = R; Co-Domain = Z x = smallest integer greater than x Ex: 3.2 = 4; -2.5 =-2

≤ +

Floor and Ceiling Functions The floor and ceiling functions map the real numbers onto the integers (RZ). The floor function assigns to rR the largest zZ with z  r, denoted by r. Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4 The ceiling function assigns to rR the smallest zZ with z  r, denoted by r. Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3 9/16/2018

Some important functions Factorial function: Domain = Range = N Error on range n! = n (n-1)(n-2) …, 3 x 2 x 1 Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120 Note: 0! = 1 by convention.

Some important functions Mod (or remainder): Domain = N x N+ = {(m,n)| m N, n  N+ } Co-domain Range = N m mod n = m - m/n n Ex: 8 mod 3 = 8 - 8/3 3 = 2 57 mod 12 = 9; Note: This function computes the remainder when m is divided by n. The name of this function is an abbreviation of m modulo n, where modulus means with respect to a modulus (size) of n, which is defined to be the remainder when m is divided by n. Note also that this function is an example in which the domain of the function is a 2-tuple.

Some important functions: Exponential Function Domain = R+ x R = {(a,x)| a  R+, x  R } Co-domain Range = R+ f(x) = a x Note: a is a positive constant; x varies. Ex: f(n) = a n = a x a …, x a (n times) How do we define f(x) if x is not a positive integer?

Some important functions: Exponential function How do we define f(x) if x is not a positive integer? Important properties of exponential functions: a (x+y) = ax ay; (2) a 1 = a (3) a 0 = 1 See:

We get: By similar arguments: Note: This determines ax for all x rational. x is irrational by continuity (we’ll skip “details”).

Some important functions: Logarithm Function Logarithm base a: Domain = R+ x R = {(a,x)| a  R+, a>1, x  R } Co-domain Range = R y = log a (x)  ay = x Ex: log 2 (8) =3; log 2 (16) =3; 3 < log 2 (15) <4. Key properties of the log function (they follow from those for exponential): log a (1)=0 (because a0 =1) log a (a)=1 (because a1 =a) log a (xy) = log a (x) + log a (x) (similar arguments) log a (xr) = r log a (x) log a (1/x) = - log a (x) (note 1/x = x-1) log b (x) = log a (x) / log a (b)

Logarithm Functions log 2 (1/4)= - log 2 (4)= - 2. Examples: log 2 (1/4)= - log 2 (4)= - 2. log 2 (-4) undefined log 2 (210 35 )= log 2 (210) + log 2 (35 )=10 log 2 (2) + 5log 2 (3 )= = 10 + 5 log 2 (3 )

Limit Properties of Log Function As x gets large, log(x) grows without bound. But x grows MUCH faster than log(x)…more soon on growth rates.

Some important functions: Polynomials Polynomial function: Domain = usually R Co-domain Range = usually R Pn(x) = anxn + an-1xn-1 + … + a1x1 + a0 n, a nonnegative integer is the degree of the polynomial; an 0 (so that the term anxn actually appears) (an, an-1, …, a1, a0) are the coefficients of the polynomial. Ex: y = P1(x) = a1x1 + a0 linear function y = P2(x) = a2x2 + a1x1 + a0 quadratic polynomial or function

We’ll talk more about growth rates in the next module…. Exponentials grow MUCH faster than polynomials: We’ll talk more about growth rates in the next module….