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) = x2 – 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: S1 S2 …… Sn T that associates with each ordered n-tuple of elements (s1, s2, …, sn) – 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 s1 and s2 of S with f(s1) = f(s2) and then show that s1 = s2 – 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) = x3 – surjective, injective, bijective
• The function f: N N where f(x) = x2 – not surjective, injective, not bijective
CS 130 – Discrete Structures 47 In General
CS 130 – Discrete Structures 48 Composition of Functions
• Let fS : T and gT : U . Then the composition function, g f , is a function from S to U defined by ( gfs )() gfs (()) . – 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) = x2 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) = 3x2, g(x) = 5x
CS 130 – Discrete Structures 50 More Examples
• Given the following function, decide whether it is 1-to-1 or onto: • f: NN, f(x) = x + 1
but 1-to-1 Proof that f is one-to-one.
Let f(s1) = f(s2) for s1, s2 . Then
s1 + 1 = s2 + 1 definition of f s1 = s2 algebra CS 130 – Discrete Structures f is one-to-one 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 is
• Show that f g = iT
CS 130 – Discrete Structures 53 Inverse Functions
• Definition: Let f: S T. If there exists a function g: T S, such that
– g f = iS and f g = iT – 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: ST and g: TU. 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 Meaning function 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. domain Starting set for a function codomain Ending set for a function image Point that results from a mapping preimage Starting point for a mapping range Collection of all images of a domain onto (surjective) Range is the whole codomain; every codomain element has a preimage one-to-one (injective) No two elements in a domain map to the same place bijection One-to-one and onto identity function Maps each element of set to itself inverse function 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: ZN where f(x) = x2 + 1 b. g: NQ 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: NN where g(x) = 2x f. h: R2 R2 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: PZ 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: NN be defined by f(x) = x + 1. Let g: NN be defined by g(x) = 3x. Calculate the following: a. (gf)(x) b. (fg)(x) c. (ff)(x) d. (gg)(x)
CS 130 – Discrete Structures 60 Exercises
• The following functions map R to R. Give an equation describing the composition functions (gf) and (fg) in each case. a. f(x) = 6x3, g(x) = 2x b. f(x) = (x-1)/2, g(x) = 4x3 c. f(x) = x , g(x) = x
CS 130 – Discrete Structures 61 Exercises
• For each of the following bijections f: RR, and g: R2 R2 find f-1 and g-1. a. f(x) = 2x b. f(x) = x3 c. f(x) = (x+4)/3 d. g(x, y) = (2x, y+1)
CS 130 – Discrete Structures 62