이산수학(Discrete Mathematics) - cs.kangwon.ac.krysmoon/courses/2011_1/dm/08.pdf · valued function f, which assigns to each number x R a particular value y=f(x), where y R. ((여러분이대수학에서이미익숙한실수집합에대한함수로이해할수있다.)

이산수학이산수학(Discrete Mathematics) (Discrete Mathematics) 이산수학이산수학(Discrete Mathematics) (Discrete Mathematics) 함수함수 (Functions)(Functions)함수함수 ( )( )

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함수의함수의 정의정의 (1/2)(1/2)1.8 Functions

From calculus, you are familiar with the concept of a real-

valued function f which assigns to each number xR a valued function f, which assigns to each number xR a

particular value y=f(x), where yR.(여러 분이 대수학에서 이미 익숙한 실수 집합에 대한 함수로 이해할 수 있다 )(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.p g g y y(함수에 대한 표기는 비단 실수 집합이 아닌 임의의 집합들을 대상으로 일반화시

킬 수 있다.)

Discrete Mathematicsby Yang-Sae MoonPage 2

1.8 Functions함수의함수의 정의정의 (2/2)(2/2)

Definition: For any sets A, B, we say that a function f from (or

“mapping”) A to B (f:AB) is a particular assignment of exactly mapping ) A to B (f:AB) is a particular assignment of exactly

one element f(x)B to each element xA.

(함수 f는 A의 원소 각각에 대해서 B의 원소를 단 하나만 대응시킴)(함수 f는 A의 원소 각각에 대해서 B의 원소를 단 하나만 대응시킴)

Definition: A partial function f assigns zero or one elements of

B to each element xA.

(부분함수 f는 A의 일부(or 전체) 원소 각각을 B의 원소 단 하나에 대응시킴)


Graphical RepresentationsGraphical Representations1.8 Functions

Functions can be represented graphically in several ways:

f A B

• •a b

f

f

•••

••

a b•

•••

y

A B x

PlotBipartite Graph

Like Venn diagrams


Examples of Functions (We’ve seen so far)Examples of Functions (We’ve seen so far)1.8 Functions

A proposition can be viewed as a function from “situations” to truth values {T,F} (명제 함수){ , } (명 )

• A logic system called situation theory.

• p=“It is raining ”; s=our situation here now• p= It is raining. ; s=our situation here, now

• p(s){T,F}

A propositional operator can be viewed as a function from ordered A propositional operator can be viewed as a function from ordered pairs of truth values to truth values: (명제 연산자)

• ((F T)) = T• ((F,T)) = T

• →((T,F)) = F

A t t h b i d 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}

Function TerminologiesFunction Terminologies1.8 Functions

If f:AB, and f(a)=b (where aA & bB), then:

A i th d i f f [정의역]• 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]p g f [원상]

(In general, b may have more than 1 pre-image.)

Th R B f f i {b | f( ) b } [치역]• The range RB of f is {b | a f(a)=b }. [치역]


Range versus Codomain (Range versus Codomain (치역과치역과 공역공역) (1/2)) (1/2)1.8 Functions

The range of a function might not be its whole codomain.(함수의 치역은 전체 공역이 아닐 수(다를 수) 있다 )(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.)


Range versus Codomain (Range versus Codomain (치역과치역과 공역공역) (2/2)) (2/2)1.8 Functions

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 f는 {A B C D E}의 성적을 부여하는 함수라 하자 )(함수 f는 {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!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• Suppose the grades turn out all As and Bs.(공부들을 잘해서 모두 A와 B를 주었다고 하자.)

• Then the range of f is {A B} but its codomain is still {A B C D E} 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.)


Function Operators (Example)Function Operators (Example)1.8 Functions

, × (“plus”, “times”) are binary operators over R. (Normal addition & multiplication.)( p )

If f,g:RR, then the followings hold: (정의)

• (f g):RR, where (f g)(x) = f(x) g(x)

• (f×g):RR, where (f × g)(x) = f(x) × g(x)(f g) , (f g)( ) f( ) g( )

Example (예제 4)

• f,g:RR, f(x) = x2, g(x) = x – x2

• (f g)(x) = f(x) g(x) = (x2) (x – x2) = x(f g)( ) f( ) g( ) ( ) ( )

• (f×g)(x) = f(x)×g(x) = (x2)×(x – x2) = x3 – x4


Images of Sets under Functions Images of Sets under Functions (집합에 대한 상)1.8 Functions

Given f:AB, and SA,

The image of S under f is simply the set of all images of The image of S under f is simply the set of all images of the elements of S. (S의 image는 S의 모든 원소의 image의 집합)

f(S) = {f(s) | sS} = {b | sS(f(s)=b)}.

Note the range of f can be defined as simply the image of f’s domain!(f의 치역은 f의 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}


f(S) {1, 4}

OneOne--toto--One Functions (One Functions (단사함수단사함수))1.8 Functions

A function is one-to-one (1-1) or injective iff every element of its range has only 1 pre-image.g y p g(치역의 모든 원소는 오직 하나의 역상(pre-image)를 가진다.)

Formally: given f:AB,“x is injective” = (x y: xy f(x)f(y))x is injective (x,y: xy f(x)f(y)).

Only one element of the domain is mapped to any given y pp y gone element of the range.(정의역의 한 원소는 치역의 한 원소에 대응)

• Domain & range have same cardinality. What about codomain?g y( may be larger)


OneOne--toto--One IllustrationOne Illustration1.8 Functions

Bipartite (2-part) graph representations of functions that are (or not) one-to-one:( )

•••

••

••

••

••

••

•• •

••• •

••

•• •

•

•

•One-to-one

•

Not one-to-one•

Not even a function!function!


Sufficient Conditions for 1Sufficient Conditions for 1--1ness1ness1.8 Functions

For functions f over numbers,

• f is strictly (or monotonically) increasing iff x>y f(x)>f(y) for all • 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 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. x3

• Converse is not necessarily true. E.g. 1/x


Onto Functions (Onto Functions (전사함수전사함수))1.8 Functions

A function f:AB is onto or surjective iff its range is equal to its codomain (bB, aA: f(a)=b). (치역과 공역이 동일하다.)( , f( ) ) ( )

An onto function maps the set A onto (over, covering) the entirety of the set B, not just over a piece of it.

3 2E.g., for domain & codomain R, x3 is onto, whereas x2

isn’t. (Why not?)


Onto IllustrationOnto Illustration1.8 Functions

Some functions that are or are not onto their codomains:

•

•••

••

•••

••

•••

••

•••

••

••

•• •

•

•• •

• •• •

• ••

Onto(but not 1-1)

Not Onto(or 1-1)

Both 1-1and onto

1-1 butnot onto


Bijection (Bijection (전단사함수전단사함수))1.8 Functions

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.)

•••

••

•• •

•

both 1-1 and onto bijection


Identity Functions (Identity Functions (항등함수항등함수))1.8 Functions

For any domain A, the identity function I:AA (or IA) is the unique function such that aA: I(a)=a.q ( )(항등함수는 정의역의 각 원소(a)를 자기 자신(I(a)=a)에 대응시키는 함수이다.

Some identity functions we’ve seen:

• ing 0, ·ing by 1g , g y

• ing with T, ing with F

• ing with ing with U• ing with , ing with U.

Note that the identity function is both one-to-one and yonto (bijective). (모든 원소에 대해 자기 자신을 대응시키므로 당연!)


Identity Function IllustrationIdentity Function Illustration1.8 Functions

••

••

•y

••

••

Domain and rangex


Composite Operator (Composite Operator (함수의함수의 합성합성) (1/2)) (1/2)1.8 Functions

For functions g:AB and f:BC, there is a special

t ll d (“ ”)operator called compose (“◦”).

• It composes (creates) a new function out of f, g by applying f to the

result of g. (함수 g의 결과에 대해 함수 f를 적용한다.)

• (f◦g):AC, where (f◦g)(a) = f(g(a)). 정의(f g) , (f g)( ) f(g( ))

• Note g(a)B, so f(g(a)) is defined and C.

N h (lik C i b lik ) i i • Note that ◦ (like Cartesian , but unlike +, , ) is non-commuting.

(Generally, f◦g g◦f.) (일반적으로, 교환법칙은 성립하지 않는다.)


Composite Operator (Composite Operator (함수의함수의 합성합성) (2/2)) (2/2)1.8 Functions

예제

• f,g:NN, 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+11


Composition IllustrationComposition Illustration1.8 Functions

(f◦g)(a)

g(a) f(b)

a g(a) = b f(g(a))g f

A B C

g f

ff◦g


Inverse Functions (Inverse Functions (역함수역함수) (1/2)) (1/2)1.8 Functions

For bijections f:AB, there exists an inverse of f, written

f1 B A hi h i th i f ti h th t f1 f If1:BA, which is the unique function such that f1◦f=I.(전단사함수 f에서 f(a) = b가 성립할 때, 함수 f의 역함수 f1는 B의 모든 원소 b

에 대해 A의 원소 를 대응시키는 함수이다 )에 대해 A의 원소 a를 대응시키는 함수이다.)

f(a)f( )

a = f-1(b) b = f(a)f-1(b)

fA B

f

f-1


Inverse Functions (Inverse Functions (역함수역함수) (2/2)) (2/2)1.8 Functions

예제

• f:ZZ, 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 – 1.


Graphs of Functions (1/2)Graphs of Functions (1/2)1.8 Functions

We can represent a function f:AB as a set of ordered

i {( f( ))| A} ll d hpairs {(a, f(a))| aA} called a graph.

(함수 f의 그래프는 (a, f(a))로 구성되는 순서쌍(pair)의 집합이다.)

Note that a, there is only 1 pair (a, f(a)).

F f ti b t d d 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.)


Graphs of Functions (2/2)Graphs of Functions (2/2)1.8 Functions

예제

2• f:ZZ, f(x) = x2

• graph of f = {…, (-2,4), (-1,1), (0,0), (1,1), (2,4), …)

●●

●●

●

●●

●

●


A Couple of Key FunctionsA Couple of Key Functions1.8 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 x에 대해서, x와 같거나 x보다 작은 수 중에서 x에 가장 가까운 정수)(실수 x에 대해서, x와 같거나 x보다 작은 수 중에서 x에 가장 가까운 정수)

• x (“ceiling of x”) is the smallest (most negative) integer x.

(실수 x에 대해서, x와 같거나 x보다 큰 수 중에서 x에 가장 가까운 정수)( , )

예제

• 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 = -3


Visualizing Floor & Ceiling FunctionsVisualizing Floor & Ceiling Functions1.8 Functions

Real numbers “fall to their floor” or “rise to their ceiling.”

Note that if xZNote that if xZ,

• x x (1.5 = 2 1.5 = 1)

31 6=2

• x x (1.5 = 1 1.5 = 2)

Note that if xZ, 1

2 ...

1.6

1.6 2

1.6=1

• x = x = x (2 = 2 = 2)

Note that if xR,

0

1 ..1.4

1.4= 1

Note that if xR,

• x = x (1.5 = 2 = 1.5)

x x ( 1 5 1 1 5)

2

3

... .

1.4= 2

1.4

3 3= 3= 3• x = x (1.5 = 1 = 1.5) 3=3= 3


Plots with Floor/Ceiling: ExamplePlots with Floor/Ceiling: Example1.8 Functions

Plot of graph of function f(x) = x/3:

Set of points (x, f(x)) +2

x+33

2


Bits versus BytesBits versus Bytes1.8 Functions

How many bytes are necessary to store 100 bits?

• 1 byte = 8 bits• 1 byte = 8 bits

• 100/8 = 12.5 = 13 bytes


Homework #2Homework #21.8 Functions


Documents

이산수학(Discrete Mathematics) - cs.kangwon.ac.krysmoon/courses/2011_1/dm/08.pdf · valued function f, which assigns to each number x R a particular value y=f(x), where y R. ((여러분이대수학에서이미익숙한실수집합에대한함수로이해할수있다.)