강원대학교 컴퓨터과학전공 문양세 이산수학 (Discrete Mathematics)  함수 (Functions)

Discrete Mathematics by Yang-Sae Moon Page 2 함수의 정의 (1/2) From calculus, you are familiar with the concept of a real- valued function f, which assigns to each number x  R a particular value y=f(x), where y  R. ( 여러 분이 대수학에서 이미 익숙한 실수 집합에 대한 함수로 이해할 수 있다.) But, the notion of a function can also be generalized to the concept of assigning elements of any set to elements of any set. ( 함수에 대한 표기는 비단 실수 집합이 아닌 임의의 집합들을 대상으로 일반화시 킬 수 있다.) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 3 Definition: For any sets A, B, we say that a function f from (or “mapping”) A to B (f:A  B) is a particular assignment of exactly one element f(x)  B to each element x  A. ( 함수 f 는 A 의 원소 각각에 대해서 B 의 원소를 단 하나만 대응시킴 ) Definition: A partial function f assigns zero or one elements of B to each element x  A. ( 부분함수 f 는 A 의 일부 (or 전체 ) 원소 각각을 B 의 원소 단 하나에 대응시킴 ) 1.8 Functions 함수의 정의 (2/2)

Discrete Mathematics by Yang-Sae Moon Page 4 Graphical Representations Functions can be represented graphically in several ways: 1.8 Functions A B a b f f x y Plot Bipartite Graph Like Venn diagrams A B

Discrete Mathematics by Yang-Sae Moon Page 5 Examples of Functions (We’ve seen so far) A proposition can be viewed as a function from “situations” to truth values {T,F} ( 명제 함수 ) A logic system called situation theory. p=“It is raining.”; s=our situation here, now p(s)  {T,F} A propositional operator can be viewed as a function from ordered pairs of truth values to truth values: ( 명제 연산자 )  ((F,T)) = T → ((T,F)) = F A set operator such as , ,  can be viewed as a function from pairs of sets to sets. ( 집합 연산자 ) Example:  ({1,3},{3,4}) = {3} 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 6 Function Terminologies If f:A  B, and f(a)=b (where a  A & b  B), then: A is the domain of f.[ 정의역 ] B is the codomain of f.[ 공역 ] b is the image of a under f.[ 상 ] a is a pre-image of b under f.[ 원상 ] (In general, b may have more than 1 pre-image.) The range R  B of f is {b |  a f(a)=b }.[ 치역 ] 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 7 Range versus Codomain ( 치역과 공역 ) (1/2) The range of a function might not be its whole codomain. ( 함수의 치역은 전체 공역이 아닐 수 ( 다를 수 ) 있다.) The range is the particular set of values in the codomain that the function actually maps elements of the domain to. ( 치역은 공역의 부분집합으로서, 함수에 의해 실제 매핑이 일어난 원소의 집합이다.) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 8 Range versus Codomain ( 치역과 공역 ) (2/2) Example Suppose I declare to you that: “f is a function mapping students in this class to the set of grades {A, B, C, D, E}.” ( 함수 f 는 {A, B, C, D, E} 의 성적을 부여하는 함수라 하자.) At this point, you know f’s codomain is: {A, B, C, D, E}, and its range is unknown!. ( 성적을 부여하기 전에 공역은 {A, B, C, D, E} 로 알 수 있으나, 그 치역은 알지 못한다.) Suppose the grades turn out all As and Bs. ( 공부들을 잘해서 모두 A 와 B 를 주었다고 하자.) Then the range of f is {A, B}, but its codomain is still {A, B, C, D, E}. ( 치역은 {A, B} 로 결정 되었지만, 공역은 그대로 {A, B, C, D, E} 이다.) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 9 Function Operators (Example) , × (“plus”, “times”) are binary operators over R. (Normal addition & multiplication.) If f,g:R  R, then the followings hold: ( 정의 ) (f  g):R  R, where (f  g)(x) = f(x)  g(x) (f×g):R  R, where (f × g)(x) = f(x) × g(x) Example ( 예제 4) f,g:R  R, f(x) = x 2, g(x) = x – x 2 (f  g)(x) = f(x)  g(x) = (x 2 )  (x – x 2 ) = x (f×g)(x) = f(x)×g(x) = (x 2 )×(x – x 2 ) = x 3 – x Functions

Discrete Mathematics by Yang-Sae Moon Page 10 Images of Sets under Functions Images of Sets under Functions ( 집합에 대한 상 ) Given f:A  B, and S  A, The image of S under f is simply the set of all images of the elements of S. (S 의 image 는 S 의 모든 원소의 image 의 집합 ) f(S)= {f(s) | s  S} = {b |  s  S(f(s)=b)}. Note the range of f can be defined as simply the image of f’s domain! (f 의 치역은 f 의 정의역에 대한 상 (image) 로 정의할 수 있다.) Example ( 예제 5) A = {a, b, c, d, e}, B = {1, 2, 3, 4}, S = {b, c, d} f(a) = 2, f(b) = 1, f(c) = 4, f(d) = 1, f(e) = 1  f(S) = {1, 4} 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 11 One-to-One Functions ( 단사함수 ) A function is one-to-one (1-1) or injective iff every element of its range has only 1 pre-image. ( 치역의 모든 원소는 오직 하나의 역상 (pre-image) 를 가진다.) Formally: given f:A  B, “x is injective” = (  x,y: x  y  f(x)  f(y)). Only one element of the domain is mapped to any given one element of the range. ( 정의역의 한 원소는 치역의 한 원소에 대응 ) Domain & range have same cardinality. What about codomain? (  may be larger) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 12 One-to-One Illustration Bipartite (2-part) graph representations of functions that are (or not) one-to-one: 1.8 Functions One-to-one Not one-to-one Not even a function!

Discrete Mathematics by Yang-Sae Moon Page 13 Sufficient Conditions for 1-1ness For functions f over numbers, f is strictly (or monotonically) increasing iff x>y  f(x)>f(y) for all x,y in domain; (strictly increasing function, 단조 증가 함수 ) f is strictly (or monotonically) decreasing iff x>y  f(x)<f(y) for all x,y in domain; (strictly decreasing function, 단조 감소 함수 ) If f is either strictly increasing or strictly decreasing, then f is one-to-one. E.g. x 3 Converse is not necessarily true. E.g. 1/x 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 14 Onto Functions ( 전사함수 ) A function f:A  B is onto or surjective iff its range is equal to its codomain (  b  B,  a  A: f(a)=b). ( 치역과 공역이 동일하다.) An onto function maps the set A onto (over, covering) the entirety of the set B, not just over a piece of it. E.g., for domain & codomain R, x 3 is onto, whereas x 2 isn’t. (Why not?) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 15 Onto Illustration Some functions that are or are not onto their codomains: 1.8 Functions Onto (but not 1-1) Not Onto (or 1-1) Both 1-1 and onto 1-1 but not onto

Discrete Mathematics by Yang-Sae Moon Page 16 Bijection ( 전단사함수 ) A function f is a one-to-one correspondence, or a bijection, or reversible, or invertible, iff it is both one-to-one and onto. ( 전사함수이면서 단사함수이면, 이를 전단사함수라 한다.) 1.8 Functions both 1-1 and onto  bijection

Discrete Mathematics by Yang-Sae Moon Page 17 Identity Functions ( 항등함수 ) For any domain A, the identity function I:A  A (or I A ) is the unique function such that  a  A: I(a)=a. ( 항등함수는 정의역의 각 원소 (a) 를 자기 자신 (I(a)=a) 에 대응시키는 함수이다. Some identity functions we’ve seen:  ing 0, ·ing by 1  ing with T,  ing with F  ing with ,  ing with U. Note that the identity function is both one-to-one and onto (bijective). ( 모든 원소에 대해 자기 자신을 대응시키므로 당연 !) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 18 Identity Function Illustration 1.8 Functions Domain and range x y

Discrete Mathematics by Yang-Sae Moon Page 19 Composite Operator ( 함수의 합성 ) (1/2) For functions g:A  B and f:B  C, there is a special operator called compose (“ ◦ ”). It composes (creates) a new function out of f, g by applying f to the result of g. ( 함수 g 의 결과에 대해 함수 f 를 적용한다.) (f ◦ g):A  C, where (f ◦ g)(a) = f(g(a)).  정의 Note g(a)  B, so f(g(a)) is defined and  C. Note that ◦ (like Cartesian , but unlike +, ,  ) is non-commuting. (Generally, f ◦ g  g ◦ f.) ( 일반적으로, 교환법칙은 성립하지 않는다.) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 20 Composite Operator ( 함수의 합성 ) (2/2) Example ( 예제 18) f,g:N  N, f(x) = 2x+3, g(x) = 3x+2 (f ◦ g)(x) = f(g(x)) = 2(3x+2)+3 = 6x+7 (g ◦ f)(x) = g(f(x)) = 3(2x+3)+2 = 6x Functions 

Discrete Mathematics by Yang-Sae Moon Page 21 Composition Illustration 1.8 Functions f◦gf◦g (f ◦ g)(a) g(a)g(a)f(b)f(b) ABC ag(a) = bf(g(a)) gf

Discrete Mathematics by Yang-Sae Moon Page 22 Inverse Functions ( 역함수 ) (1/2) For bijections f:A  B, there exists an inverse of f, written f  1 :B  A, which is the unique function such that f  1 ◦ f=I. ( 전단사함수 f 에서 f(a) = b 가 성립할 때, 함수 f 의 역함수 f  1 는 B 의 모든 원소 b 에 대해 A 의 원소 a 를 대응시키는 함수이다.) 1.8 Functions f(a)f(a) AB a = f -1 (b) b = f(a) f -1 (b) f f -1

Discrete Mathematics by Yang-Sae Moon Page 23 Inverse Functions ( 역함수 ) (2/2) Example ( 예제 15) f:Z  Z, f(x) = x + 1 f is a strictly increasing function, and thus, f is bijection. Let f(x) = y, then y = x + 1 and x = y – 1. Therefore, f -1 (y) = y – Functions

Discrete Mathematics by Yang-Sae Moon Page 24 Graphs of Functions (1/2) We can represent a function f:A  B as a set of ordered pairs {(a, f(a))| a  A} called a graph. ( 함수 f 의 그래프는 (a, f(a)) 로 구성되는 순서쌍 (pair) 의 집합이다.) Note that  a, there is only 1 pair (a, f(a)). For functions over numbers, we can represent an ordered pair (x, y) as a point on a plane. A function is then drawn as a curve (set of points) with only one y for each x. ( 수에 대한 함수인 경우, 함수를 평면상에 그릴 수 있다.) 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 25 Graphs of Functions (2/2) Example ( 예제 20) f:Z  Z, f(x) = x 2 graph of f = {…, (-2,4), (-1,1), (0,0), (1,1), (2,4), …) 1.8 Functions ● ● ● ● ● ● ●

Discrete Mathematics by Yang-Sae Moon Page 26 A Couple of Key Functions In discrete math, we will frequently use the following functions over real numbers:  x  (“floor of x”) is the largest (most positive) integer  x. ( 실수 x 에 대해서, x 와 같거나 x 보다 작은 수 중에서 x 에 가장 가까운 정수 )  x  (“ceiling of x”) is the smallest (most negative) integer  x. ( 실수 x 에 대해서, x 와 같거나 x 보다 큰 수 중에서 x 에 가장 가까운 정수 ) Example ( 예제 21)  1/2  = 0,  1/2  = 1,  3.1  = 3,  3.1  = 4,  7  = 7,  7  = 7  -1/2  = -1,  -1/2  = 0,  -3.1  = -4,  -3.1  = Functions

Discrete Mathematics by Yang-Sae Moon Page 27 Visualizing Floor & Ceiling Functions Real numbers “fall to their floor” or “rise to their ceiling.” Note that if x  Z,  x     x  (  1.5  =  2    1.5  =  1)  x     x  (  1.5  =  1    1.5  =  2) Note that if x  Z,  x  =  x  = x (  2  = 2 =  2  ) Note that if x  R,  x  =   x  (  1.5  =  2 =   1.5  )  x  =   x  (  1.5  =  1 =   1.5  ) 1.8 Functions 0 1 22 3  1.6  =2  1.4  =  2  1.4  1.4  =  1  1.6  =1 33  3  =  3  =  3

Discrete Mathematics by Yang-Sae Moon Page 28 Plots with Floor/Ceiling: Example Plot of graph of function f(x) =  x/3  : ( 그림 10) 1.8 Functions x Set of points (x, f(x)) +3 22 +2 33

Discrete Mathematics by Yang-Sae Moon Page 29 Bits versus Bytes How many bytes are necessary to store 100 bits? 1 byte = 8 bits  100/8  =  12.5  = 13 bytes 1.8 Functions

Discrete Mathematics by Yang-Sae Moon Page 30 Homework #2 $1.6 의 연습문제 : 2(b), 8(e), 24(c) $1.7 의 연습문제 : 4, 8, 14(b) $1.8 의 연습문제 : 8(e, g), 18(b), 22(b, c) Due Date: 1.8 Functions