2011-2012 Lecture 2.2 3.1 Functions Sequences Relations CH3

8/3/2019 2011-2012 Lecture 2.2 3.1 Functions Sequences Relations CH3

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Algorithms and DataStructures (ET3155)

Lecture 2

Functions, Sequence and Relations(Chapter 3)

Said Hamdioui Carlo Galuzzi

Computer Engineering Lab

Delft University of Technology

2011-2012

S.Hamdioiui, CE, EWI, TUDelft ET3155 Algorithms and Data Structures, 2011-20122

Previous lecture

Methods of proofs

Direct proof

Contradiction proof (Indirect proof)

Proof by contrapositive

Proof by cases

Proof by logical equivalence

Proof by induction


Goals

Functions Explain the concept of a function and its main characteristics

Prove if a function has some properties (e.g., one-to-one,

onto, etc)

Sequences Explain the concept of a sequence and be able to manipulate

them

Relations Explain the concept of Relation and its main properties

Prove if a relation has some properties

Prove that a relation is an equivalence relation

Use a matrix to describe a Relation and to prove its properties


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Topics

Functions

Sequences and Strings

Relations

Matrices


Topics

Functions


Relations


Functions (1)

A functionf from X to Y (in symbols f : X Y) assigns to eachmember of X exactly one memberof Y; i.e., if two pairs (x,y)and (x,y) f, then y = y

Domainof f = X

Rangeof f ={y| y=f(x) for some xX} [Range is a subset of Y]

To visualize a function:

Graph

Arrow diagram

Example 1:

f={(a,3),(b,3),(c,5),(d,1)} is a function

Dom(f) = X = {a, b, c, d},

Rng(f) = {1, 3, 5}

f(a) = f(b) = 3, f(c) = 5, f(d) = 1

Arrow diagram


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Functions (2) Domainof f = X

Rangeof f =

{ y | y = f(x) for some x X}

A function f : X Y assigns to each x inDom(f) = X a unique element y in Rng(f) Y.

Therefore, no two pairs in f have thesame first coordinate.

Examples:

X={a,b,c,d}

f1={(a,3), (b,5), (c,1), (a,2),(d,3)}. Is f1 a function?

f2={(a,3),(b,2),(c,5)}. Is f2 a function?

difference with relations


Functions: Example

Let x be a integer and y a positive integer

r = x mod y is the remainder when x is divided by y

Examples:

1 = 13 mod 3

6 = 234 mod 19

4 = 2002 mod 111

mod is called the modulus operator

Modulus operator play an important role in mathand computer science


Hash functions

Hashing: a technique that allows fast searching

Element n is stored at position h(n) of a table

his called the hashing function

Example:

Table with N=11 locations 0, 1, , 10

N has to be a prime to allow probing all cells

h(n) = n mod 11

Insert 15, 558, 32, 132, 102, 5

0 1 2 3 4 5 6 7 8 9 10

132 102 15 5 558 32


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Collision Resolution

Suppose we want to insert 257

257 mod 11 = 4 but cell 4 is already occupied (with 15)

Collision has occurred

Need collision resolution policy

E.g.: find next highest unoccupied cell (0 succeeds 10)

(linear probing)

0 1 2 3 4 5 6 7 8 9 10

132 102 15 5 558 32257

0 1 2 3 4 5 6 7 8 9 10

132 102 15 5 558 32


Collision Resolution

Suppose we want to insert 257

257 mod 11 = 4 but cell 4 is already occupied (with 15)

Need collision resolution policy

E.g.: use secondary hash function h2. Probe sequence:

( h1(n) + j h2(n) ) mod N where

j=1, 2, N-1, (N is a prime)

Common choice:

h2(n) = (m - n mod m) where m


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One-to-one functions

A function f : X Y is one-to-one (orinjective) if for eachy Y, there exists at most one x X with f(x) = y.

Alternative definition: f : X Y is one-to-one if for eachpair of distinct elements x1, x2 X,if f(x1) = f(x2) then x1 = x2.

Examples: The function f(x) = 2x+1 from the set of real numbers to

itself is one-to-one

The function f : R Rdefined by f(x) = x2 is not one-to-one, since for every real number x, f(x) = f(-x).

f: N N defined by f(n) = 2n n2 is not one-to-one sincef(2) = f(4) = 0.

f={(1,a),(2,b),(3,a)} is ..?


Onto functions

A function f : X Y is ontoY (orsurjective) iffor each y Y there exists at least one x X withf(x) = y, i.e. Rng(f) = Y. (each y has an x)

Examples

1

234

a

bcd

X Yf

1

234

a

bcd

X Yf

Onto (and one-to-one) Not onto(and one-to-one)


Bijective functions

A function f : X Y is bijective f is one-to-one andonto

Examples:

A linear function f(x) = ax + b is a bijective function from the set

of real numbers to itself

The function f(x) = x3 is bijective from the set of real numbers toitself.

The function f : R Rdefined by f(x) = x2 is not bijectivebecause it is not one-to-one (for every real number x,f(x) = f(-x)).


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Inverse function

Given a function y = f(x), the inverse f-1 is the set

{(y, x) | y = f(x)}.

Example: f={(1,a),(2,b),(3,c)} f-1={(a,1),(b,2),(c,3)}

The inverse f-1 of f is not necessarily a function.

Example: if f(x) = x2, then f-1 (4) = 4 = 2, not a uniquevalue and therefore f-1 is not a function.

However, if f is a bijective function, it can beshown that f-1 is a function.


Exercises!!!

Exam Jan 08:

Given the following function: f(x)=3x -2

Define one-to-one function. Determine and proofwhether the function f(x) is one-to-one. Thedomain is a set of real numbers

Define onto function. Determine if the functionf(x) is onto function.


Composition of functions

Given two functions g : X Y and f : Y Z, thecomposition f g is defined as follows:

(f g)(x) = f(g(x)) for every x X.

Example: g(x) = x2 -1, f(x) = 3x + 5. Then

(f g)(x) = f(g(x)) = f(x2 -1) = 3(x2 -1) +5 = 3x2 +2

Composition of functions is associative:

f (g h) = (f g) h,

But, in general, it is not commutative:

f g g f.


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Binary operators* A binary operatoron a set X is a function f that

associates a single element of X to every pair of

elements in X.In other words:

f : X x X X andf(x1, x2) X for every pair of elements x1, x2.

Examples:

X={1,2,3,}; f(x,y)= x+y for x,y X => f a binary operator

Binary operators are addition, subtraction andmultiplication of real numbers, taking unions orintersections of sets, concatenation of two strings over aset X, etc.


Unary operators* A unary operatoron a set X associates to each

single element of X one element of X.

In other words:

f : X X andf(x) X for every element x.

Example: Let X = U be a universal set and P(U) the power set of U.

Define f : P(U) P(U) the function defined by f (A) = Ac(i.e., the set complement of A in U) for every A U.

Then f defines a unary operator on P(U).


Summary (1)

Functions

Definition, presentation (arrow diagram, graph)

Hash functions Collision resolution policies

One-to-one function

Onto function

Bijective function (Bijection)

Inverse of a function

Composition of functions

Binary operator versus unary operator


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Algorithms and DataStructures (ET3155)

Lecture 3

Functions, Sequence and Relations(Chapter 3)

Said Hamdioui

Computer Engineering Lab

Delft University of Technology

2010-2011S.Hamdioiui, CE, EWI,TUDelft

ET3155 Algorithms and Data Structures,2011-2012


Goals Functions

Explain the concept of a function and its main characteristics

Prove if a function has some properties (e.g., one-to-one,onto, etc)

Sequences Explain the concept of a sequence and be able to manipulate

them

Relations Explain the concept of Relation and its main properties

Prove if a relation has some properties

Prove that a relation is an equivalence relation

Use a matrix to describe a Relation and to prove its properties


Topics

Functions


Relations


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Sequences and strings

A sequenceis an ordered list of numbers, usually

defined according to a formula: sn = f(n) for n = 1, 2, 3,...

E.g., Cn=1+0.5(n-1) for n=1,2,3,4

If s is a sequence {sn| n = 1, 2, 3,},

s1 denotes the first element,

s2 the second element,

sn the nth element

{n} is called the indexing set/ domainof the sequence.

Usually the indexing set is N (natural numbers) or an

infinite subset of N.

nis the index of the sequence


Examples of sequences

Let s = {sn} be the sequence defined by sn = 1/n , for n = 1, 2, 3, First few elements are: 1, , 1/3, , 1/5,1/6,

Let s = {sn} be the sequence defined by sn = n

2 + 1, for n = 1, 2, 3, First few elements are: 2, 5, 10, 17, 26, 37, 50,

Fibonacci sequence: f1 = 1, f2 = 1, fn = fn-1 + fn-2 for n = 2, 3, 4,

First few elements: f1 = f2 =1, f3 = 2, f4 = 3, f5 = 5, f6 = 8

Recurrence relation, more later


Increasing and decreasing

A sequence s = {sn} is said to be

(strictly) increasingif sn < sn+1

nondecreasingif sn < sn+1

(strictly) decreasingif sn > sn+1,

nonincreasingif sn > sn+1,

for every n = 1, 2, 3,

Examples: Sn = 4 2n, n = 1, 2, 3, is decreasing: {2, 0, -2, -4, -6,}

Sn = 2n -1, n = 1, 2, 3, is increasing: {1, 3, 5, 7, 9, }

Sn = n/2, n = 1, 2, 3, is nondecreasing: {0, 1, 1, 2, 2, 3, }


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Subsequences

A subsequenceof a sequence s = {sn} is asequence t = {tn} that consists of certain

elements ofsretained in the original ordertheyhad in s

Example:

let s = {sn = n | n = 1, 2, 3,}

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,

Let t = {tn = 2n | n = 1, 2, 3,}

2, 4, 6, 8, 10, 12, 14, 16,

t is a subsequence ofs

Let m={4,3,2,1}

m is not a subsequent of s (order not maintained)


Sum notation If {an} is a sequence, then the sum

n

ai = am + am+1 + + ani = m

This is called the sum orsigma notation,

where the Greek letter indicates a sum ofterms from the sequence

iis called the index

mis the lower limit

nis the upper limit


Product notation

If {an} is a sequence, then the product

n

ai = am am+1ani=m

This is called the product notation, where the

Greek letter (pi) indicates a product ofterms of the sequence


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Strings

Let X be a nonempty set. A string over Xis afinite sequence of elements from X.

Example: if X = {a, b, c}

Then = bbaccc is a string over X

Notation: bbaccc = b2ac3

The length of a string is the number of elements of and is denoted by ||. If = b2ac3 then || = 6.

The null stringis the string with no elements andis denoted by the Greek letter. || = 0.


More on strings (1)

X* = {all strings over X including }

X+ = X* - {}, the set of all non-null strings E.g., X={a,b}

Some elements of X* are: , a, ab, abab, b2a6ba

Concatenationof two strings and is theoperation on strings consisting of writing followed

by to produce a new string Example: = bbaccc and = caaba,

then = bbaccccaaba = b2ac4a2ba

|| = | | + ||


More on strings (2)

Bit strings are strings over {0,1}

Substring of: is obtained by selecting some of allconsecutive elements of

Definition

A string is a substring of the string if there are strings and with =


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Exercises (Exam Jun 08)

Exam Jan 08

Given the following sequence:

zn=(2+n).3n; n 0

Find a formula for zn-1 ?

Find a formula for zn-2 ?

Prove that {zn} satisfies:

zn = 6 zn-1 9 zn-2 for n 2


Topics

Functions


Relations

Equivalence relations

Matrices and relations

Relational database


Relations

Given two sets X and Y, the Cartesian productXxYis the set of all ordered pairs (x,y) where xX andyY XxY = {(x, y) | xX and yY}

E.g., X={1,2}, Y={a,b}; XxY={(1,a), (1,b), (2,a), (2,b)}

A (binary) relationRfrom a set X to a set Y is asubset of the Cartesian product XxY

If (x,y) R, we write xRy (x is related to y)

Example: X = {1, 2, 3} and Y = {a, b}

R= {(1,a), (1,b), (2,b), (3,a)} is a relation from X to Y


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Domain and range

Given a relation Rfrom X to Y,

The domainofRis the set Dom(R) = { xX | (x, y) Rforsome yY}

The rangeofRis the set Rng(R) = { yY | (x, y) Rforsome x X}

Example: if X = {1, 2, 3} and Y = {a, b, c}

R= {(1,a), (1,b), (2,b)}

Then: Dom(R)= {1, 2}, Rng(R) = {a, b}


Example of a relation

Let X = {1, 2, 3} and Y = {a, b, c, d}.

Define R= {(1,a), (1,d), (2,a), (2,b), (2,c)}

The relation can be pictured by a graph

What is the domain?

What is the range?


Properties of relations (1)

Let Rbe a relation on a set X (i.e. Ris a subset of XxX)

Ris reflexiveif (x,x) Rfor every xX Examples

- R={(a,a), (b,c), (c,b), (d,d)} on X={a,b,c,d} is not reflexive

- R on X={1,2,3,4} defined by (x,y) R if x < y is reflexive

Ris symmetricif for all x, y X, if (x,y)Rthen (y,x) R Examples

- R={(a,a), (b,c), (c,b), (d,d)} on X={a,b,c,d} is symmetric

- R on X={1,2,3,4} defined by (x,y) R if x < y is notsymmetric


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Properties of relations (2)Let Rbe a relation on a set X (i.e. Ris a subset of XxX)

Ris antisymmetric

if for all x,yX, if (x,y)R and (y,x)R, then x=y. Examples

- R={(a,a), (b,c), (c,b), (d,d)} on X={a,b,c,d} is not antisymmetric

- R on X={1,2,3,4} defined by (x,y) R if x < y is antisymmetric

Ris transitive

if (x,y) Rand (y,z) Rimply (x,z)R Examples

- R={(a,a), (b,c), (c,b), (d,d)} is not transitive

- R on X={1,2,3,4} defined by (x,y) R if x < y is transitive


Examples of relations

x divides y is a relation on lN

{(1,1),(1,2),(1,3),,

(2,2),(2,4),(2,6),,

(3,3),(3,6),(3,9),}

- Reflexive? Symmetric? Transitive?

x < y is a relation on lN

{(1,1),(1,2),(1,3),,(2,2),(2,3),(2,4),,

(3,3),(3,4),(3,5),}

- Reflexive? Symmetric? Transitive?


Order relations

Let X be a set and Ra relation on X

Ris a partial orderon X if R is reflexive,antisymmetric and transitive.

Suppose R is partial order, let x,yX If (x,y) or (y,x) are in R, then x and y are comparable

If (x,y) Rand (y,x) Rthen x and y are incomparable

Ifevery pair of elements in X are comparable, thenRis a total orderon X

x divides yhas someincomp. el.

eitherx


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Partial Order

Is on natural numbers a partial order?

is reflexive because n n

is transitive because if l n and n m then lm

is antisymmetric if m n and n m then n=m

is a total order?

yes, because for every n and m: n m or m n

Is < on natural numbers partial order?

No because it is not reflexive : n


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Equivalence relationsLet X be a set and Ra relation on X

Ris an

equivalence relationon X if

Ris

reflexive, symmetric and transitive.

Example:R={(1,1),(1,3),(1,5),(2,2),(2,4),(3,1),(3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)} on {1,2,3,4,5}

R is reflexive because (1,1)(2,2) etc. are in R

R is symmetric because both (x,y) and (y,x) are in R

R is transitive because when (x,y) and (y,z) are in Rthen (x,z) is also in R

R is an equivalence relation

Note: directed graphs can be used


Equivalence relation

Examples:

Is = on natural numbers an equivalencerelation ?

= is reflexive : n=n = is symmetric : if m=n then n=m = is transitive : if m=n and n=l then m=l

R is neither reflexive nor transitive

Is R={(a,a), (b,c), (c,b), (d,d)} onX={a,b,c,d} an equivalent relation?


Partitions

A partitionSon a set X is a collection {A1, A2,, An}of subsets of X, such that

A1A2A3An = X

Aj Ak = for every j, k with j k, 1 < j, k < n.

Example:

if X = {integers}, E = {even integers) and O = {odd integers}

then S= {E, O} is a partition of X.

If X={1,2,3,4,5,6,7,8}, thenS = {{1,4,5}, {2,6}, {3,7}, {8}} is a partition of X

S = {{1,2,4,5}, {2, 3, 6}, {7}, {8}} is NOT a partition of X


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Partitions and equivalence relations

Theorem: Let Sbe a partition on a set X.

Define a relation Ron X by xRy to mean that for some subset

T Sboth x, y belongs to T.Then: Ris an equivalence relationon X. an equivalence relation on a set X corresponds to apartition of X and conversely.

Proof:

By definition of partition, x belongs to T; thus xRx (reflexive)

Suppose xRy. Then both x and y belong to same subset T. Sodo y and x, yRx (symmetric).

Suppose xRy and yRz. x and y belong to same subset A andy and z belong to same subset T. Since y belongs to exactlyone memberS, we must have T=A. Hence xRz (transitive).


Partitions and equivalence relations

Consider partition S={ {1,3,5} ,{2,6},{4}} ofX={1,2,3,4,5,6}

Consider the relation R on X defined by xRy if x, ybelongs to the same subset TS (theorem of the previousslide)

R = {(1,1)(1,3)(1,5)(3,1)(3,3)(3,5)(5,1)(5,3)(5,5)}

is an equivalence relation

because {1,3,5} is in S.

The complete relation is :{(1,1)(1,3)(1,5)(3,1)(3,3)(3,5)(5,1)(5,3)(5,5)(2,2)(2,6)(6,2)(6,6)(4,4) }


Equivalence classes

Let X be a set and let Rbe an equivalence relation on X.Let a X.

Define [a] ={ xX | xRa }(i.e., the set of all elements in X that are related to a)

Let S = { [a] | a X }( i.e., a collection of subset of [a])

Theorem: S is a partition on X. [proof: see the book]

The sets [a] are called equivalence classesof X given bythe relation R.

Given a, b X, then [a] = [b] or [a][b] =


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Partitions and equivalence relations Consider the equivalence relation R on

X={1,2,3,4,5,6} defined as:

R={(1,1)(1,3)(1,5)(3,1)(3,3)(3,5)(5,1)(5,3)(5,5)(2,2)(2,6)(6,2)(6,6)(4,4)

}

Equivalence class [1] consists of all x such that (x,1)R: [1]={1,3,5}=[3]=[5]

The other equivalence classes are : [2]=[6]={2,6}

[4]={4}

Hence S = { [a] | aX} [Theorem previous slide]={{1,3,5},{2,6},{4}} is a partition


Partitions and equivalence relations Example:

let X={1,2,...,10}. xRy means 3 divides x-y; Ris an equivalence relation

members of equivalence classes :

equivalence=has same remainder when divided by 3

[1]={x X| 3 divides x-1}={xX | xR1} {1,4,7,10}

[2]={2,5,8}

[3]={3,6,9}

[1]=[4]=[7]=[10]

[2]=[5]=[8]

[3]=[6]=[9]

Therefore S = { [a] | aX}= {{1,4,7,10},{8,5,2}{3,6,9}}is a partition


Exercises

Exercise 1: How many equivalence relations are there on the set {1,2,3}}?

Exercise 2:

Let X={1, 2, , 10}

Define a relation R on XxX by (a,b)R(c,d) if a+d=b+c

Is R an equivalence relation on XxX?

Exercise 3: Exam Jan 10


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Matrices of Relations

Let X, Y be sets and Ra relation from X to Y

Matrix of the relation R (say A = (aij) ) isdefined as follows:

Rows of A = elements of X

Columns of A = elements of Y

Element ai,j = 0 if the element of X in row i and

the element of Y in column j are not related

Element ai,j = 1 if the element of X in row i and

the element of Y in column j are related


The matrix of a relation (1) Example:

Let X = {1, 2, 3}, Y = {a, b, c, d}

Let R= {(1,a), (1,d), (2,a), (2,b), (2,c)}

The matrix A of the relation Ris

Obviously the matrix depends on the ordering of X and Y

a b c d

1 1 0 0 1

2 1 1 1 0

3 0 0 0 0

A =

X

Y


The matrix of a relation (2)

IfRis a relation from a set X to itself and A is thematrix ofRthen A is a square matrix.

Example: Let X = {a, b, c, d} and R= {(a,a), (b,b),(c,c), (d,d), b,c), (c,b)}. Then

1000

0110

0110

0001

d

c

b

a

A


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The matrix of a relation on a set X

Let A be the square matrix of a relation R from Xto itself. Let A2 = the matrix product AA.

Ris reflexive All terms aii in the maindiagonal of A are 1

Ris symmetric aij = aji for all i and j, i.e. R is a symmetric relation on X if A is a symmetric

matrix

Ris transitive whenevercij in C = A2 isnonzero then entry aij in A is also nonzero.


Example

R = {(a,a),(b,b},{b,c},(c,b),(c,c),(d,d)}

1000

0220

0220

0001

1000

0110

0110

0001

2A

A

R is symmetric

R is reflexive

R is transitive

R is an equivalence relation


Relational databases

A binaryrelation Ris a relation among twosets X and Y, already defined as R X x Y.

An n-aryrelation Ris a relation among n setsX1, X2,, Xn, i.e. a subset of the Cartesian

product, R X1 x X2 xx Xn. Thus, Ris a set of n-tuples (x1, x2,, xn) where xk

Xk, 1 < k < n.


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Databases

A databaseis a collection of records that are

manipulated by a computer. They can beconsidered as nsets X1 through Xn, each of whichcontains a list of items with information.

Database management systemsare programsthat help access and manipulate information

stored in databases.


Relational database model Columns of an n-ary relation are called attributes

E.g., Attribute Age with a domain: all positive integers lessthan 100

An attribute is a keyifno two entries have the samevalue e.g. ID is a key but Age is NOT (ID is unique but age is not)

A queryis a request for information from the database E.g. find all persons with age 50

EmployeeID Name Age

22 Jan 22

71 Mark 50

101 Peter 50


Operators

The selection operatorchooses n-tuples from a relation bygiving conditions on the attributes

E.g., Employee [Age=50] will select (71,Mark,50) and (101,Peter,50)

The projection operatorchooses two or more columns and

eliminates duplicates

E.g., Employee [ID, Name] will select (22,Jan), (71,Mark) and (101, Peter)

Thejoin operatormanipulates two relations

form the Cartesian product of two relations and select those tuplessatisfying a certain condition

E.g., merge each two rows of two

different tables when a certain

condition is satisfied

Employee

ID Name Age

22 Jan 22

71 Mark 50

101 Peter 50


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SQL - Structured Query Language*

SELECT

SELECT Name, AgeFROM Employee

WHERE Age>=60

JOINSELECT employees.name

FROM employees

JOIN invoices

ON employees.ID=invoices.employeeID

SQL tutorial can be found at

http://www.tizag.com/sqlTutorial


Summary (2)


Definition of a sequence and a string

Subsequence versus substring

Increasing versus decreasing sequences

Nonincreasing versus nondecreasing sequences

Sum/Sigma notation versus product notation

Null string

Let X be a nonempty finite set X*: the set of all strings over X, including the null string

X+: the set of all nonnull strings over X

Concatenation of strings


Summary (3)

Binary relations, domain, range

Relation properties Reflexive, Symmetric, Antisymmetric, Transitive, Partial order

Inverse relation

Equivalence relations and partitions

Equivalence classes

Matrix of a relation Reflexive, symmetric and transitive

Binary relation versus n-ary relation

Relational database (attribute, key, query)

Operators (selection, projection, join)


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Suggested Exercises

From Chapter 3 Self-test 1, 4, 9, 14, 17

Documents

2011-2012 Lecture 2.2 3.1 Functions Sequences Relations CH3