discrete structure for computer science

8/13/2019 discrete structure for computer science

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What is a set?A Set is any well-definedcollection of objects called the elements or members

of the set.

Well-definedmeans that it is possible to describe if a given object belongs to

the collection or not.

Describing a Set

Way one: List the elements of the set between braces (f ini te elements)

e.g. the set of all positive integers that are less than 4 : {1, 2, 3}

Way two: Specify a property that the elements of the set have in common

e.g. R={x | x is a real number }

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1.1. Sets and Subset

Property of the elements


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The order of the Set

{1, 2, 3}={1, 3, 2}={2, 3, 1}={2, 1,3 }={3, 1, 2}={3, 2, 1}

Repeated elements can be ignored{1, 2, 3, 1} = {1, 2, 3}

Several commonly used sets

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The relationships between Element & Set

Usually, we use uppercase letters such as A, B and C to denote

sets, and lowercase letters such as a, b, c, x, y and z to denote the

elements of sets

Binary cases: for a given element x and set A1: x belongs to A denoted by x A

2: x does not belong to A denoted by x A

Fuzzy Sets

The collections of richpeople, young girls, so on and so forthNote: The words rich, young, beautiful , cool, hot, fat, thin etc. are

fuzzy (not well def ined).

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SubsetIf every element of A is also an element of B, namely, if whether x A then x

B, we say that A is a subsetof B, denoted by A B . Otherwise, .

Venn diagrams

A B

4


B A BA A B


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5


A B

A=B: A B & B A A

U

An universal set (U) is a set containing all objects for

which the discussion is meaningful.


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The cardinality of a finite setA set A is called finite if it has ndistinctelements, where n N.

In this case, n is the cardinality of A and is denoted by |A|.

e.g. A={1,2,3,1} |A| = 3

B={a, b, c, d, e, a}, |B| = 5

|A| < |B|

A set that is not finite is called infinite, for instances, N, Z, Q, R

as mentioned in Example 3.

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Power set of a set A

If A is a set, then the set of all subsets of A is called the power

set of A and is denoted by P(A).

e.g. A={1,2,3}

Then P(A) consists of the following subsets of A: {}, {1}, {2},{3}, {1,2}, {1,3}, {2,3}, and {1,2,3}

|P(A)| = 2^n, why? Assuming n = |A| N


8


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UnionIf A and B are sets, we define their union as the set consisting of

all elements that belong to A or B and denote it by A U B.

A U B ={ x | x A or x B}

1.2. Operations on Sets

9

A

U

B


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Intersection

If A and B are sets, we define their intersection as the set

consisting of all elements that belong to both A and B and

denoted it by A B.

A B ={ x | x A and x B}


10

U

A B


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Complement of B with respect to AIf A and B are two sets, we define the complement of B with

respect to A as the set of all elements that belong to A but not to

B, and we denote it by A - B

A - B ={ x | x A and x B}


11

U

AA B


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ComplementIf U is a universal set containing A, then U-A is called the

complement of A and is denoted by

= {x | x A}


12

U

A


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Symmetric differenceIf A and B are two sets, we define their symmetric difference as

the set of all elements that belong to A or to B, but not to both A

and B, and we denote it by A B

A B = {x | (x A and x B) or (x B and x A) }


13

U

A B


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Commutative PropertiesA U B = B U A ; A B = B A

Associative Properties

A U (B U C) = ( A U B ) U C

A (B C) = ( A B ) C

Distribution Properties

A (B U C) = ( A B ) U ( A C )

A U (B C) = ( A U B ) ( A U C )

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Idempotent Properties

A U A =A ; A A = A

Properties of the complement

De Morgans Law

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Properties of a Universal setA U U = U

A U = A

Properties of the empty set

A U = A

A =

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How to proof above properties?e.g. Proof:

Proof: suppose x , then we have x AB, so

x or ,x which means that x . Thus,

Conversely, suppose x then we have x A or x B ,

so x A B, which means that x .Thus

Therefore,

A common style of proof for statements about sets is to choose an

element in one of the sets and see what we know about it.17



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Addition PrincipleTheorem 2: If A and B are finite sets, then

|A U B| = |A| + |B| - |A B |

18


U

A B

A B


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Example 9A computer company wants to hire 25 programmers to handle

systems programming jobs and 40 programmers for applications

programming. Of those hired, 10 will be expected to perform

jobs of both types. How many programmers must be hired? (at

least? )Solution:

A: the set of system programmers hired

B: the set of applications programmers hired, then

|A| = 25, |B| = 40, |A B| =10

|A U B| = |A| + |B| - |A B |

= 25 + 40 -10 =55


19


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Generalized case for three setsTheorem 3: Let A, B and C be finite sets. Then

|A U B U C| = |A| + |B| + |C| - |AB| - |BC|-|AC| + |ABC|

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A B

C

AB

BCAC

ABC


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Sequences

a list of objects arranged in a definite order

finite or infinite, explicit or recursive

some elements may be repeated in a sequence

set corresponding to a sequenceset of all distinctelements in the sequence;

A set is called countable if it is the set corresponding

to some sequence All finite sets are countable Not all infinite sets are countable (e.g. the set of all real

numbers)

AU, characteristic function fAof A is defined foreach xU,

fA (x)=1 if xA; fA(x)=0 if xA

fAB= fAfB fAB= fA+fB-fAfB

fAB= fA+fB-2fAfB


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While sets have no order and no repeated elements,sequenceshave order and can contain repeats at differingpositions in the order.

The set {5,2,5} = {5,2} = {2,5} The sequence (5,2,5) (5,2) (2,5)

Actually, there is a notion of a multisetor bagthat wesometimes use. It has no order, but repeated elements areallowed. Since position is irrelevant, we just record eachunique elements with a count.

We can talk about the k-th elementof a sequence, but not ofa set or multiset.

Finite sequences are often called tuples. Those of length kare k-tuples. A 2-tuple is also called apair.

Computer representation of Sets and Subsets


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Computer representation of Sets and Subsets

Let U={a,b,e,g,h,r,s,w}. The array of length 8 shown below represents U.

1, if x= a,b,e,g,h,r,s,w

fU(x)=

0 otherwise

If S={a,e,r,w} then

1, if x= a,e,r,w

fS(x)=0 if x= b,g,h,s

The array representation of subset S is

1 0 1 0 0 1 0 1


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Strings and Regular Expressions

alphabet A: a set of symbols string: a finitesequence of symbols in A

A*consists of all finitestrings from A

empty string : contains no symbols

catenation of w1and w2is w1 w2

a regular expressionover A is a stringconstructed from the elements of A and thesymbols (, ), , *, , according to the followingdefinition: (a recursive definition)

RE1. The symbol is a regular expression.

RE2. IfxA, the symbolxis a regular expression. RE1 and RE2 provide initial regular expressions


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Strings and Regular Expressions (cont)

RE3. If and are regular expressions, then the

expression is regular.RE4. If and are regular expressions, then the

expression () is regular.

RE5.If is a regular expression, then the expression

()*is regular. each regular expression corresponds to a regular

subsetsof A*, or regular setif no reference to A isneeded

the concept of regular expressions is important forthe study of the syntax of programming languages

To compute the regular set corresponding to aregular expression, use the following correspondencerules:


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Example


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Example

Ex 1. Let A=(a,b,c,..,z}, the usual English alphabet. Then A* consists of

ordinary words, such as ape,queen and so on as well as yxaloble,esy etc.All

finite sequences from are A*, whether they have meaning or not.

Ex 2. If w1=s1s2snand w2=t1t2tnare the elements of A* for some set A wedefine the catenation of w1and w2as the sequence s1s2snt1t2t2. The

catenation of w1and w2is written as w1.w2or w1w2and its another element

of A*.

Ex 3. If A={Ram,swims,Sam, walks, sleep,slowly} then A* contain real

sentences like Ram walks slowly as well as nonsense sentences likeswim sleep walks

Ex 4. Let A={0,1} 0*(0V1)* is regular expression.

By RE2, 0 and 1 are regular expression. Thus (0V1) us a regular expression

by RE4 and so 0* and (0V1)* are regular by RE5 and finally we see that

0*(0V1)* is regular by RE3.


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Associated with each regular expression over A, there is a corresponding

subset of A*. Such sets are called regular subsets of A* or just regular sets.

Ex 1. Let A={a,b,c}. Then the regular expression a* corresponds to the set of all

finite sequences of as, such as aaa,aaaaa and so on. The regular

expression a(bVc) corresponds to the set {ab,ac} A* The regularexpression ab(bc)*corresponds to the set of all strings that begin with ab and

then repeat the symbol bc n times where n>=0. This set includes ab,abbc,

abbcbc and so on.

Ex 2. If A={0,1} the regular set of corresponding to regular expression

00*(0V1)*1 corresponds to the set of all sequences of 0s and 1s that beginwith at least one 0 and end with at least one 1.

M t i


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Matrix

An array of numbers arranged in mhorizontal rowsand nvertical columns:

A = a11 a12 a13 . a1na21 a22 a23 a2n

am1 am2 am3 amn

The ith row of A is [ai1, ai2, ai3, ain]; 1 im

Thejth column of A is a1ja2j ; 1 jna3jamj


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We say that A is a matrix mx n. If m= n, then A

is a square matrix of order n, and a11, a22, a33,

..annform the main diagonal of A.

aijwhich is in the ith row andjth column, is said

to be the i ,jth element of Aor the (i , j) entry

of A, often written as A = [aij].


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Ex 2:

A = 8 0 0 0

0 3 0 0

0 0 7 00 0 0 1

A square matrix A = [aij], for which every entryoff the main diagonal is zero, that is aij= 0 for i

j, is called a diagonal matrix.


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Two mx nmatrices A and B, A = [aij] and B =

[bij], are said to be equalif aij= bijfor 1 im,

1 jn; that is, if corresponding elements are

the same.

Ex 3:

A = a 5 3 B = 1 5 x

2 7 -1 y 7 -1

3 b 0 3 4 0

So, if A = B, then a = 1, x = 3, y = 2, b = 4.

Matrix summation


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Matrix summation

If A = [aij] and B = [bij] are mx nmatrices, then

the sum of A and B is matrix C = [cij], defined by

cij= a

ij+ b

ij; 1 i m, 1 j n.

C is obtained by adding the corresponding

elements of A and B.


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A = 1 5 3 B = 2 0 3

2 7 -1 6 1 3

3 4 0 -3 1 9

C = 3 5 6

8 8 4

0 5 9

The sum of the matrices A and B is defined onlywhenA and B have the same number of rows and the samenumber of columns (same dimension).


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A matrix in when all of its entries are zero is

called zero matrix, denoted by 0.

Theorems involved in summation :

A + B = B + A.

(A + B) + C = A + (B + C).A + 0 = 0 + A = A.


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Matrices Product

If A = [aij] is an mxpmatrix and B = [bij] is apx

nmatrix, then the product of A and B, denotedAB, will produce the mx nmatrix C = [cij],

defined by

cij= ai1b1j+ ai2b2j+ + aipbpj;1 in, 1 j m

That is, elements ai1, ai2, .. aipfrom ith row of A

and elements b1j, b2j, .. bpjfromjth column of B,

are multiplied for each corresponding entries

and add all the products.


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Ex 5:

A = 2 3 -4 B = 3 1

1 2 3 -2 2

5 -3

2 x 3 3 x 2

AB = 2(3) + 3(-2) + -4(5) 2(1) + 3(2) + -4(-3)

1(3) + 2(-2) + 3(5) 1(1) + 2(2) + 3(-3)


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= 6620 2 + 6 + 12

34 + 15 1 + 49

= -20 20

14 -42 x 2


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If A is an mxp matrix and B is apx nmatrix, in

which AB will produce mx n, BA might be

produce or not depends on:

nm, then BA cannot be produced.

n= m,pm@ n, then we can get BA but the

size will be different from AB. n = m=p, A B, then we can get BA, the size

of BA and AB is the same, but AB BA.

n= m=p, A = B, then we can get BA, the sizeof BA and AB is the same, and AB = BA.

Identity matrix


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Identity matrix

Let say A is a diagonal matrix nx n. If all entries on its

diagonal are 1, it is called identity matrix, ordered n,

written as I. Ex 7:

1 0 1 0 0 1 0 0 0

0 1 0 1 0 0 1 0 0

0 0 1 0 0 1 0

0 0 0 1

Theorems involved are: A(BC) = (AB)C.

A(B + C) = AB + AC.

(A + B)C = AC + BC.

IA = AI = A.

Transposition Matrix


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Transposition Matrix If A = [aij] is an mx nmatrix, then A

T= [aij]Tis a nx

mmatrix, where

aijT= aji; 1 i m, 1 jn

It is called transposition matrixfor A.

Ex 8:

A = 2 -3 5 AT= 2 6

6 1 3 -3 1

5 3

Theorems involved are: (AT)T= A

(A + B)T= AT+ BT

(AB)T= BTAT


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Matrix A = [aij] is said to be symmetricif AT= A, that

is aij= aji,

A is said to be symmetric if all entries aresymmetrical to its main diagonal.

Ex 9:

A = 1 2 -3 B = 1 2 -3

2 4 5 2 4 0

-3 5 6 3 2 1

Symmetric Not Symmetric, why?

Boolean Matrix and Its Operations


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Boolean Matrix and Its Operations

Boolean matrixis an mx nmatrix where all of its entriesare either 1 or 0 only.

There are three operations on Boolean:

a)Join by

Given A = [aij] and B = [bij] are Boolean matrices with thesame dimension,jo in byA and B, written as A B, will

produce a matrix C = [cij], wherecij= 1 if aij= 1 OR bij= 1

0 if aij= 0 AND bij= 0

b)MeetMeetfor A and B, both with the same dimension, writtenas A B, will produce matrix D = [dij] where

dij= 1 if aij= 1 AND bij= 1

0 if aij= 0 OR bij= 0


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Ex 10: A =1 0 1 B = 1 1 0

0 1 1 0 0 11 1 0 0 1 0

0 1 0 1 1 0

A B= 1 1 1 A B= 1 0 00 1 1 0 0 1

1 1 0 0 1 0

1 1 0 0 1 0


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c) Boolean product

If A = [aij] is an mxpBoolean matrix, and B = [bij] is apx nBoolean matrix, we can get a Boolean productfor A and B written as A B, producing C, where:

cij= 1 if aik= 1 AND bkj= 1; 1 kp.

0 other than that

It is using the same way as normal matrix product.

+

+

E 11


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Ex 11:

A = 1 0 0 0 B = 1 1 0

0 1 1 0 0 1 0

1 0 1 1 1 1 03 x 4 0 0 1

4 x 3

A B = 1 + 0 + 0 + 0 1 + 0 + 0 + 0 0 + 0 + 0 + 0

0 + 0 + 1 + 0 0 + 1 + 1 + 0 0 + 0 + 0 + 01 + 0 + 1 + 0 1 + 0 + 1 + 0 0 + 0 + 0 + 1

A B = 1 1 0

1 1 01 1 1

3 x 3

+


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Exercise 3: A = 1 0 0 0 B = 0 1 0 0 C = 0 0 1 0

0 1 1 0 0 0 1 1 1 0 0 00 0 0 1 0 1 0 1 1 1 0 01 1 0 0 0 0 1 0 1 1 1 0

Find:a) A Bb) A Bc) A Bd) A Ce) A Cf) A Cg) B Ch) B Ci) B C

+

+

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