Section 4.4 Functions CS 130 Discrete Structures
Function Definitions Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first component of an ordered pair. S is the domain of the function. T is the codomain of the function. If (s, t) belongs to the function, then t is denoted by f(s) t is the image of s under f s is a preimage of t under f f is said to map s to t CS 130 Discrete Structures 39
Diagrams Representation If A and B are finite sets, an arrow diagram shows a function f from A to B by drawing an arrow from each element in A to the corresponding element of B Two properties must be held in the arrow diagram according to the definition of function: Every element of A has an arrow coming out of it No one element of A has two arrows coming out of it that point to two different elements of B Example CS 130 Discrete Structures 40
Examples Which of the following are functions? f: S T where S = T = {1, 2, 3}, f = {(1,1),(2,3),(2,1)} g: Z N where g(x) = x (absolute value of x) h: N N where h(x) = x 4 f: R R where f(x) = 4x 1 more examples from the book For f: Z Z where f(x) = x 2 what is the image of 4 what are the preimages of 9 CS 130 Discrete Structures 41
Several Common Math Functions Floor function associates with each real number x the greatest integer less than or equal to x E.g., 2.8 =?, - 2.8 =? Ceiling function associates with each real number x the smallest integer greater than or equal to x E.g, 2.8 =?, -2.8 =? Modulo function f(x) = x mod n associates with x the remainder when x is divided by n we can write x = qn+ r, r is between [0, n-1] 10 mod 3 = 1 CS 130 Discrete Structures 42
Functions With Multiple Variables A function can include more than one variable. A function can be defined as f: S 1 S 2 S n T that associates with each ordered n-tuple of elements (s 1, s 2,, s n ) Example: f: Z Z Z is given by f(x,y) = x+y CS 130 Discrete Structures 43
Properties of Functions: Surjective Three properties: surjective (onto), injective, bijective Let f: S T be an arbitrary function every member of S has an image under f and all the images are members of T the set R of all such images is called the range of the function f A function f: S T is an onto, or surjective, function if the range of f equals the codomain of f CS 130 Discrete Structures 44
Example of Surjective Functions To prove a function to be surjective: need to show that an arbitrary member of the codomain T is a member of the range R, thus it is the image of some member of the domain, we have T R To disprove it: if we can find one member of the codomain that is not the image of any member of the domain Let f: Q Q where f(x) = 3x + 2 Let g: Z N where g(x) = x Let f: R R where f(x) = 4x 1 CS 130 Discrete Structures 45
Properties of Functions: Injective, Bijective A function f: S T is an one-to-one or injective, if no member of T is the image under f of two distinct elements of S To prove a function is injective: we assume that there are elements s 1 and s 2 of S with f(s 1 ) = f(s 2 ) and then show that s 1 = s 2 To disprove it: counterexample, where an element in the range has two preimages in the domain A function f: S T is bijective, if it is both surjective and injective CS 130 Discrete Structures 46
Examples The function g: R R where g(x) = x 3 surjective, injective, bijective The function f: N N where f(x) = x 2 not surjective, injective, not bijective CS 130 Discrete Structures 47
In General CS 130 Discrete Structures 48
Composition of Functions Let f : S T and g : T U. Then the composition function, g f, is a function from S to U defined by ( g f )( s) g( f ( s)). Function f is applied first, and then function g It is not always possible to take any two arbitrary functions and compose them since the domain and the ranges have to be compatible. Note that composition preserves the properties of being onto and being one-to-one Composition on two bijections is a bijection CS 130 Discrete Structures 49
Examples Let f: R R be defined by f(x) = x 2 and g: R R be defined by g(x) = x What is the value of (g f) (2.3)? g(f(2.3)) = g(5.39) = 5 What is the value of (f g) (2.3)? f(g(2.3)) = f(2) = 4 Order is important in function composition The following functions map R to R. Give an equation describing the composition functions f g and g f in each case: f(x) = 3x 2, g(x) = 5x CS 130 Discrete Structures 50
More Examples Given the following function, decide whether it is 1-to-1 or onto: f: N N, f(x) = x + 1 but 1-to-1 Proof that f is one-to-one. Let f(s 1 ) = f(s 2 ) for s 1, s 2. Then s 1 + 1 = s 2 + 1 definition of f s 1 = s 2 algebra f is one-to-one CS 130 Discrete Structures 51
Proofs Let f: S T and g: T U, and assume that both f and g are one-to-one (injective) functions. Prove that g f is a one-to-one function. Let f: S T and g: T U be functions Prove that if g f is 1-to-1, so is f. Prove that if g f is onto, so is g. Find an example where g f is 1-to-1, but g is not 1- to-1. Find an example where g f is onto but f is not onto. CS 130 Discrete Structures 52
Inverse Functions Bijective functions have another property: Every element s in S has an image in T and every element of T has a unique inverse image in S since the f is onto and one-to-one If there is a function f which has a one-to-one correspondence from a set S to a set T, then there is a function g from T to S that "undoes" the action of f. This function g is called the inverse function for f. Then g f (s) = s maps each element to itself. Such a function which leave an element unchanged is called the identity function on S and is denoted by i s Show that f g = i T CS 130 Discrete Structures 53
Inverse Functions Definition: Let f: S T. If there exists a function g: T S, such that g f = i S and f g = i T Then g is called the inverse function of f and is denoted by f 1 Example: f: R R given by f(x) = 3x + 4 is a bijection. Describe f 1 Theorem on Bijections and Inverse Functions: Let f: S T. Then f is a bijection if and only if f 1 exists CS 130 Discrete Structures 54
More Proofs Let f and g be bijections, f: S T and g: T U. Then f -1 and g -1 exist. Also g f is a bijection from S to U. CS 130 Discrete Structures 55
Summary of Function Terminologies Term function domain codomain image preimage range onto (surjective) one-to-one (injective) bijection identity function inverse function Meaning A function f from set S to set T is a relationship between elements of S and elements of T where each element of S is related to a unique element of T. It is denoted by f: S T. Starting set for a function Ending set for a function Point that results from a mapping Starting point for a mapping Collection of all images of a domain Range is the whole codomain; every codomain element has a preimage No two elements in a domain map to the same place One-to-one and onto Maps each element of set to itself For a bijection, a new function that maps each codomain element back where it came from CS 130 Discrete Structures 56
Exercises Let S = {0, 2, 4, 6} and T = {1, 3, 5, 7}. Determine whether each of the following sets of ordered pairs is a function with domain S and codomain T. If so, it is 1-to-1? Is it onto? a. {(0, 2), (2, 4), (4, 6), (6, 0)} b. {(6, 3), (2, 1), (0, 3), (4, 5)} c. {(2, 3), (4, 7), (0, 1), (6, 5)} d. {(2, 1), (4, 5), (6, 3)} e. {(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)} CS 130 Discrete Structures 57
Exercises Which of the following are functions from the domain to the codomain given? Which functions are 1-to-1 or onto or both? Describe the inverse function for any bijective function. a. f: Z N where f(x) = x 2 + 1 b. g: N Q where g(x) = 1/x c. h: Z x N Q where h(z, n) = z/(n+1) d. f: {1, 2, 3} {p, q, r} where f = {(1, q), (2, p), (3, r)} e. g: N N where g(x) = 2 x f. h: R 2 R 2 where h(x, y) = (y + 1, x + 1) CS 130 Discrete Structures 58
Exercises Let P be the power set of {a, b, c}. A function f: P Z follows: for A in P, f(a) = the number of elements in A. Is f 1-to-1? Is f onto? CS 130 Discrete Structures 59
Exercises Let f: N N be defined by f(x) = x + 1. Let g: N N be defined by g(x) = 3x. Calculate the following: a. (g f)(x) b. (f g)(x) c. (f f)(x) d. (g g)(x) CS 130 Discrete Structures 60
Exercises The following functions map R to R. Give an equation describing the composition functions (g f) and (f g) in each case. a. f(x) = 6x 3, g(x) = 2x b. f(x) = (x-1)/2, g(x) = 4x 3 c. f(x) = x, g(x) = x CS 130 Discrete Structures 61
Exercises For each of the following bijections f: R R, and g: R 2 R 2 find f -1 and g -1. a. f(x) = 2x b. f(x) = x 3 c. f(x) = (x+4)/3 d. g(x, y) = (2x, y+1) CS 130 Discrete Structures 62