ch2. language of mathematics

7/29/2019 ch2. Language of Mathematics

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R. Johnsonbaugh,Discrete Mathematics

5th edition, 2001

Chapter 2

The Language of Mathematics


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2.1 Sets

Set =a collection of distinct unorderedobjects

Members of a set are called elements How to determine a set

Listing:

Example: A = {1,3,5,7}

Description

Example: B = {x | x = 2k + 1, 0 < k < 3}


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Finite and infinite sets

Finitesets

Examples:

A = {1, 2, 3, 4}

B = {x | x is an integer, 1 < x < 4}

Infinitesets

Examples:

Z = {integers} = {, -3, -2, -1, 0, 1, 2, 3,} S={x| x is a real number and 1 < x < 4} = [0, 4]


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Some important sets

The emptyset has no elements.

Also called null setorvoid set.

Universalset: the set of all elements aboutwhich we make assertions.

Examples:

U = {all natural numbers} U = {all real numbers}

U = {x| x is a natural number and 1< x


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Cardinality

Cardinality of a set A (in symbols |A|) is the

number of elements in A

Examples:

If A = {1, 2, 3} then |A| = 3

If B = {x | x is a natural number and 1< x< 9}

then |B| = 9

Infinite cardinality Countable (e.g., natural numbers, integers)

Uncountable (e.g., real numbers)


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Subsets

X is a subsetof Y if every element ofX is also contained in Y

(in symbols X Y) Equality: X = Y if X Y and Y X

X is a proper subsetof Y if X Y butY X Observation: is a subset of every set


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Power set

The power setof X is the set of all subsets of X,in symbolsP(X),

i.e.P(X)= {A | A X}

Example: if X = {1, 2, 3},

thenP(X) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

Theorem 2.1.4: If |X| = n, then |P(X)| = 2n

.


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Set operations:

Union and Intersection

Given two sets X and Y

The unionof X and Y is defined as the set

X Y = { x | x X or x Y}

The intersectionof X and Y is defined as the set

X Y = { x | x X and x Y}

Two sets X and Y are disjointif X Y =


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Complement and Difference

The differenceof two sets

X Y = { x | x X and x Y}

The difference is also called the relative complement

of Y in X

Symmetric difference

X Y = (X Y) (Y X)

The complementof a set A contained in auniversal set U is the set Ac = U A

In symbols Ac = U - A


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Venn diagrams

A Venn diagram provides a graphic view ofsets

Set union, intersection, difference,

symmetric difference and complements canbe identified


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Properties of set operations (1)

Theorem 2.1.10: Let U be a universal set, and

A, B and C subsets of U. The followingproperties hold:

a) Associativity: (A B) C = A (B C)

(A B) C = A (B C)

b) Commutativity: A B = B A

A B = B A


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c) Distributive laws:

A(BC) = (AB)(AC)A(BC) = (AB)(AC)

d) Identity laws:

AU=A A = Ae) Complement laws:

AAc = U AAc =


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f) Idempotent laws:

AA = A AA = Ag) Bound laws:

AU = U A =

h) Absorption laws:A(AB) = A A(AB) = A


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i) Involution law: (Ac)c = A

j) 0/1 laws: c = U Uc =

k) De Morgans laws for sets:

(AB)c

= Ac

Bc

(AB)c = AcBc


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2.2 Sequences and strings

A sequenceis an ordered list of numbers,usually defined according to a formula: sn = a

function of n = 1, 2, 3,...

If s is a sequence {sn| n = 1, 2, 3,}, s1 denotes the first element,

s2the second element,

snthe nth element

{n} is called the indexing setof the sequence.Usually the indexing set is N (natural numbers)

or an infinite subset of N.


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Examples of sequences

Examples:

1. Let s = {sn} be the sequence defined by

sn= 1/n , for n = 1, 2, 3,

The first few elements of the sequence are: 1, , 1/3, ,

1/5,1/6,

2. Let s = {sn} be the sequence defined bysn = n

2+ 1, for n = 1, 2, 3,

The first few elements of s are: 2, 5, 10, 17, 26, 37, 50,


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Increasing and decreasing

A sequence s = {sn} is said to be

increasingif sn < sn+1

decreasingis sn > sn+1,

for every n = 1, 2, 3,Examples:

Sn = 42n, 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,


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Subsequences

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

elements of s retained in the original orderthey had in s

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

1, 2, 3, 4, 5, 6, 7, 8,

Let t = {tn= 2n | n = 1, 2, 3,} 2, 4, 6, 8, 10, 12, 14, 16,

t is a subsequence of s


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Sigma notation

If {an} is a sequence, then the sum

m

ak = a1 + a2+ + amk= 1

This is called the sigma notation, where the

Greek letter indicates a sum of termsfrom the sequence


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Pi notation

If {an} is a sequence, then the product

m

ak = a1a2amk=1

This is called the pi notation, where theGreek letterindicates a product of termsof 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 (lambda). It haslength zero.


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More on strings

Let X* = {all strings over X including }

Let X+ = X* - {}, the set of all non-null strings

Concatenationof two strings and is theoperation on strings consisting of writing followed by to produce a new string Example: = bbaccc and = caaba,

then = bbaccccaaba = b2ac4a2baClearly, || = | | + ||


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2.3 Number systems

Binarydigits: 0 and 1, called bits.

In this section we study: binary, hexadecimaland octalnumber systems.

Review ofdecimalsystem: Example: 45,238 is equal to

8 ones 8 x 1 = 8

3 tens 3 x 10 = 30

2 hundreds 2 x 100 = 200

5 thousands 5 x 1000 = 5000

4 ten thousands 4 x 10000 = 40000


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Binary number system

From binary to decimal:

The number 1101011 is equivalent to 1 one 1 x20 = 1

1 two 1x21

= 2 0 four 0x22 = 0

1 eight 1x23 = 8

0 sixteen 0x24 = 0

1 thirty-two 1x25

= 32 1 sixty-four 1x26 = 64

107 in decimal base


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From decimal to binary

The number 7310 is equivalent to 73 = 2 x 36 + remainder 1

36 = 2 x 18 + remainder 0





7310 = 10010012(write the remainders in reverse order preceded by

the quotient)


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Binary addition table

0 1

0 0 1

1 1 10


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Adding binary numbers

Example: add 1001012 + 1100112

1 1 1 carry ones

1001012

1100112

10110002


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Hexadecimal number system

Decimal system

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 A B C D E F

Hexadecimal system


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Hexadecimal to decimal

The hexadecimal number 3A0B16 is

11 x 160 = 11

0 x 161 = 0

10 x 162 = 2560

3 x 163 = 12288

1485910


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Decimal to hexadecimal

Given the number 234510

2345 = 146x16 + remainder 9146 = 9x16 + remainder 2

234510

is equivalent to the hexadecimal number 92916


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Hexadecimal addition

Add 23A16 + 8F16

23A16

+ 8F16

2C916


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2.4 Relations

Given two sets X and B, its Cartesian productXxY is the set of all ordered pairs (x,y) where

xX and yY

In symbols XxY = {(x, y) | xX and yY}

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

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

R= {(1,a), (1,b), (2,b), (3,a)} is a relation between Xand Y


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Domain and range

Given a relation Rfrom X to Y,

The domainofRis the set

Dom(R) = { xX | (x, y) Rfor some yY}

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

Example:

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

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

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


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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:


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Properties of relations

Let Rbe a relation on a set X

i.e. Ris a subset of the Cartesian product XxX

Ris reflexiveif (x,x) Rfor every xX

Ris symmetricif for all x, y X such that (x,y)Rthen (y,x) R

Ris transitiveif (x,y) Rand (y,z) Rimply

(x,z) R Ris antisymmetricif for all x,yX such that

xy, if (x,y) Rthen (y,x) R


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Order relations

Let X be a set and Ra relation on X

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

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 areincomparable

If every pair of elements in X are

comparable, then Ris atotal orderon X


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Inverse of a relation

Given a relation Rfrom X to Y, its inverse R-1is the relation from Y to X defined by

R-1 = { (y,x) | (x,y) R}

Example: ifR= {(1,a), (1,d), (2,a), (2,b), (2,c)}then R-1= {(a,1), (d,1), (a,2), (b,2), (c,2)}


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2.5 Equivalence relations

Let X be a set and Ra relation on X

Ris an equivalence relationon X Ris

reflexive, symmetric and transitive. Example: Let X = {integers} and Rbe the

relation on X defined by: xRy x - y = 5. Itis easy to show that Ris an equivalence

relation on the set of integers.


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Partitions

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

thatA1A2A3An = XAj Ak = for every j, k with j k,

1 < j, k < n.Example: if X = {integers}, E = {even

integers) and O = {odd integers}, thenS= {E, O} is a partition of X.


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Partitions and equivalence relations

Theorem 2.5.1: Let Sbea partition on a set X.

Define a relation Ron X by xRy if x, y are in thesame set T for T S. Then Ris an equivalencerelationon X.

i.e. an equivalence relation on a set X corresponds to a

partition of X and conversely.


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Equivalence classes

Let X be a set and let Rbe an equivalencerelation on X. Let a X.

Define [a] ={ xX | xRa }

Let S = { [a] | a X }

Theorem 2.5.9: S is a partition on X.

The sets [a] are called equivalence classes

of X induced by the relation R. Given a, b X, then [a] = [b] or [a][b] =


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Set of equivalence classes

IfRis an equivalence relation on a set X, defineX/R= {[a] | a X }.

Theorem 2.5.16: If each equivalence class on a

finite set X has k elements, then X/Rhas |X|/kelements, i.e. |X/R| = |X|/ k.


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2.6 Matrices of relations

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

Write the matrix A = (aij) of the relation 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 andthe element of Y in column j are related


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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

A =

a b c d

1 1 0 0 1

2 1 1 1 0

3 0 0 0 0


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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

A =

a b c d

a 1 0 0 0

b 0 1 1 0c 0 1 1 0

d 0 0 0 1


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The matrix of a relation on a set X

Let A be the square matrix of a relation Rfrom X to 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 whenever cij in C = A2 isnonzero then entry aij in A is also nonzero.


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2.7 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 nsets X1, 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 n sets X1 through Xn, each of

which contains a list of items with information.

Database management systemsare

programs that help access and manipulateinformation stored in databases.


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Relational database model

Columns of an n-ary relation are called attributes

Anattribute is a keyif no two entries have the

same value e.g. social security number

A queryis a request for information from thedatabase


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Operators

The selection operatorchooses n-tuplesfrom a relation by giving conditions on the

attributes

The projection operatorchooses two ormore columns and eliminates duplicates

Thejoin operatormanipulates tworelations


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2.8 Functions

A functionf from X to Y (insymbols f : X Y) is arelation from X to Y such that

Dom(f) = X and if two pairs(x,y) and (x,y) f, then y = y

Example:

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

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

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


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Domain and Range

Domainof f = X

Rangeof f =

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

A function f : X Y assigns toeach x in Dom(f) = X a unique

element y in Rng(f) Y.

Therefore, no two pairs in f havethe same first coordinate.


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Modulus operator

Let x be a nonnegative integer and y a positive

integer

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

by yExamples:

1 = 13 mod 3

6 = 234 mod 19

4 = 2002 mod 111

mod is called the modulus operator


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One-to-one functions

A function f : X Y is one-to-one for each y Y there exists at most one x Xwith f(x) = y.

Alternative definition: f : X Y is one-to-one for each pair of distinct elements x1, x2 X thereexist two distinct elements y1, y2 Y such thatf(x1) = y1 and f(x2) = y2.Examples:

1. The function f(x) = 2x from the set of real numbers to itself isone-to-one

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


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Onto functions

A function f : X Y is onto

for each y Y there exists at least one x Xwith f(x) = y, i.e. Rng(f) = Y. Example: The function f(x) = ex from the set of real

numbers to itself is not onto Y = the set of all real

numbers. However, if Y is restricted to Rng(f) = R +,

the set of positive real numbers, then f(x) is onto.


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Bijective functions

A function f : X Y is bijective

f is one-to-one and onto

Examples: 1. A linear function f(x) = ax + b is a bijective function from

the set of real numbers to itself

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

numbers to itself.


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Inverse function

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

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

The inverse f-1 of f is not necessarily a

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

unique value and therefore f is not a function.

However, if f is a bijective function, it can be

shown that f-1 is a function.

Exponential and


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Exponential and

logarithmic functions

Let f(x) = 2x and g(x) = log 2 x = lg x

f g(x) = f(g(x)) = f(lg x) = 2 lg x = x g f(x) = g(f(x)) = g(2x) = lg 2x = x

Therefore, the exponential and logarithmic

functions are inverses of each other.


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Composition of functions

Given two functions g : X Y and f : Y Z,the composition 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. Thenf g(x) = f(g(x)) = f(3x + 5) = (3x + 5)2 - 1

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 thatassociates a single element of X to every pair of

elements in X, i.e. f : X x X X and f(x1, x2) Xfor every pair of elements x1, x2.

Examples of binary operators are addition,

subtraction and multiplication of real numbers, takingunions or intersections of sets, concatenation of two

strings over a set X, etc.


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Unary operators

A unary operatoron a set X associates toeach single element of X one element of X.

Examples:

1. Let X = U be a universal set and P(U) the powerset of U. Define f : P(U) P(U) the functiondefined by f (A) = A', the set complement of A in U,

for every A U. Then f defines a unary operatoron P(U).


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String inverse

Let X be any set, X* the set of all strings over X.

If = x1x2xn X*, let f() = -1 = xnxn-1x2x1,the string written in reverse order.

Then f :X* X* is a function that defines a unaryoperator on X*.

Observe that -1 = -1 =

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