Functions

Discrete Structure

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Functions. Basic-Terms.DEF: A function f : A B is given by a

domain set A, a codomain set B, and a rule which for every element a of A, specifies a unique element f (a) in B. f (a) is called the image of a, while a is called the pre-image of f (a). The range (or image) of f is defined byf (A) = {f (a) | a A }.

3

Functions

Let us take a look at the function f:PC withP = {Linda, Max, Kathy, Peter}C = {Boston, New York, Hong Kong,

Moscow}

f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = New York

Here, the range of f is C.

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Functions. Basic-Terms.

f : Z R is given by f (x ) = x 2

A1: domain is Z, co-domain is RA2: image of -3 = f (-3) = 9A3: pre-images of 3: none as 3 isn’t

an integer! pre-images of 4: -2 and 2

A4: range is the set of perfect squares f (Z) = {0,1,4,9,16,25,…}

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One-to-One, Onto, Bijection. Intuitively.

Represent functions using “node and arrow” notation:One-to-One means that no clashes occur.

BAD: a clash occurred, not 1-to-1

GOOD: no clashes, is 1-to-1

Onto means that every possible output is hit BAD: 3rd output missed, not onto

GOOD: everything hit, onto

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One-to-One, Onto, Bijection. Intuitively.

Bijection means that when arrows reversed, a function results. Equivalently, that both one-to-one’ness and onto’ness occur. BAD: not 1-to-1. Reverse

over-determined:

BAD: not onto. Reverseunder-determined:

GOOD: Bijection. Reverseis a function:

8

Properties of Functions

Is f injective?Yes.Is f surjective?Yes.Is f bijective?Yes.

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston

New YorkNew York

Hong KongHong Kong

MoscowMoscow

LLüübeckbeckHelenaHelena

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One-to-One, Onto, Bijection. Formal

Definition.DEF: A function f : A B is:

one-to-one (or injective) if different elements of A always result in different images in B. onto (or surjective) if every element in B is hit by f. I.e., f (A ) = B.a one-to-one correspondence (or a bijection, or invertible) if f is both one-to-one as well as onto. If f is invertible, its inverse f -1 : B A is well defined by taking the unique element in the pre-image of b, for each b B.

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One-to-One, Onto, Bijection. Examples.

1. f : Z R, f (x ) = x 2: none2. f : Z Z, f (x ) = 2x : 1-13. f : R R, f (x ) = x 3: 1-1, onto,

bijection, inverse is f (x ) = x (1/3)

4. f : Z N, f (x ) = |x |: onto5. f (x ) = the father of x : none

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CompositionWhen a function f spits out elements of

the same kind that another function g eats, f and g may be composed by letting g immediately eat each output of f.

DEF: Suppose that g : A B and f : B C are functions. Then the composite f g : A C is defined by setting

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

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Composition. Examples.

Q: Compute g f where 1. f : Z R, f (x ) = x 2

and g : R R, g (x ) = x 3

2. f : Z Z, f (x ) = x + 1and g = f -1 so g (x ) = x – 1

3. f : {people} {people},f (x ) = the father of x, and g = f

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Composition. Examples.1. f : Z R, f (x ) = x 2

and g : R R, g (x ) = x 3

f g : Z R , f g (x ) = x 6

2. f : Z Z, f (x ) = x + 1and g = f -1

f g (x ) = x (true for any function composed with its inverse)

3. f : {people} {people},f (x ) = g(x ) = the father of x

f g (x ) = grandfather of x from father’s side

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Composition

Example:

f(x) = 7x – 4, g(x) = 3x,f:RR, g:RR

(fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101

(fg)(x) = f(g(x)) = f(3x) = 21x - 4

Ceiling and FloorDEF: Given a real number x : The

floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integer greater or equal to x.

NOTATION: floor(x) = x , ceiling(x) = x

Q: Compute 1.7, -1.7, 1.7, -1.7.

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Ceiling and Floor

A: 1.7 = 1, -1.7 = -2, 1.7 = 2, -1.7 = -1

Q: What’s the difference between the floor function and the (int) casting function in Java?

a0

a1

a2

anan-1

a0=0a1=2a2=6a3=12a4=20

Recursive Definitions

F(0) = 0; F(n + 1) = F(n) + 1;

F(0) = 1; F(n + 1) = 2 F(n);

F(0) = 1; F(n + 1) = 2F(n)

The First-Order Linear Recurrence Relation

There are many sequences that satisfy a a nn n 1 3 0, .

For example, 5,15,45,135,... or 7,21,63,189,.... To pinpointthe particular sequence described, we need to know one of theterms of the sequence. (boundary condition, or initial conditionsince usually a0 is specified)

a a n an n 1 03 0 5, ,

The general solution of the recurrence relation is a constant, and is unique and is given by

+

0

a dan d a A

a Ad n

n n

nn

10

0

,, ,

, .

The First-Order Linear Recurrence Relation

nonhomogeneous linear recurrence relation

Ex. 10.4 time complexity of bubble sort algorithm

an=an-1+(n-1), n>1, a1=0, where an=the number of comparisons to sort n numbers

an- an-1= n-1an-1- an-2= n-2an-2- an-3= n-3

a2- a1= 1+

an =1+2+3+...+(n-1)=(n2-n)/2

The Second-Order Linear Homogeneous Recurrence Relation with Constant Coefficients

Be careful not to draw conclusions from a few (or even,perhaps, many) particular instances.Ex. 10.14 Arrange pennies contiguously in each row where eachpenny above the bottom row touches two pennies in the row below it.

a1=1,a2=1,a3=2,a4=3,a5=5,a6=8,... Is an=Fn? NO

SequencesDEF: Given a set S, an (infinite) sequence in S

is a function N S. Symbolically, a sequence is represented using

the subscript notation ai . This gives a way of specifying formulaically

Note: Other sets can be taken as ordering models.

Q: Give the first 5 terms of the sequence defined by the formula

)2

πcos( iai

Sequence ExamplesA: Plug in for i in sequence 0, 1, 2, 3, 4:

Formulas for sequences often represent patterns in the sequence.

Q: Provide a simple formula for each sequence:

a) 3,6,11,18,27,38,51, … b) 0,2,8,26,80,242,728,…c) 1,1,2,3,5,8,13,21,34,…

1,0,1,0,1 43210 aaaaa

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Sequence ExamplesA: Try to find the patterns between numbers.a) 3,6,11,18,27,38,51, … a1=6=3+3, a2=11=6+5, a3=18=11+7, … and in

general ai +1 = ai +(2i +3). This is actually a good enough formula. Later we’ll learn techniques that show how to get the more explicit formula:

ai = 6 + 4(i –1) + (i –1)2

b) 0,2,8,26,80,242,728,… If you add 1 you’ll see the pattern more clearly.

ai = 3i –1c) 1,1,2,3,5,8,13,21,34,…This is the famous Fibonacci sequence given by

ai +1 = ai + ai-1

Definition of Finite Automaton

The finite automaton has 5-tuple (Q, ∑, δ,

q0, F)

Q is a set of states

∑ is a set of input alphabet

δ is a transition function or rule for moving

q0 is a start state

F is a set of accept states

Finite automaton

If the input is: 0101010101 & 0100 & 110000 & 0101000000 & 101000What is the outputs?

q3q2q1 1

0 10

0, 1

The finite automaton M is (Q, ∑, δ, q0, F)

Where Q = {q1, q2, q3} ∑ = {0, 1}

q0 is the state q1 F = {q2}

q3q2q1 1

0 10

0, 1

δ is describes as

0 1q1 q1 q2q2 q3 q2q3 q2 q2

q3q2q1 1

00

0, 1

Example

r2

q1

sba

b a

r1

q2

b

a

a a b

b

For the start state, is even because it is

possible to input 0 so far

qevenqodd

10

1

0

For the accept state, we consider qodd

because we want to test odd numbers

qodd

Example

The inputs a, b, baba, baa, bbb,

q2

q1a

q3

bb

a

a, b

00,1

00

1

1

1

The machine accepts a string if the process ends in a double circle

Anatomy of a Deterministic Finite Automaton

states

states

q0

q1

q2

q3start state (q0)

accept states (F)

Anatomy of a Deterministic Finite Automaton

00,1

00

1

1

1

q0

q1

q2

q3

The alphabet of a finite automaton is the set where the symbols come from:

The language of a finite automaton is the set of strings that it accepts

{0,1}

Q = {q0, q1, q2, q3}

Σ = {0,1}

: Q Σ → Q transition function*q0 Q is start state

F = {q1, q2} Q accept states

M = (Q, Σ, , q0, F) where

0 1

q0 q0 q1

q1 q2 q2

q2 q3 q2

q3 q0 q2

*

q2

00,1

00

1

1

1

q0

q1

q3

M