sets, sequences, matrix

Sets and Subsets Operations on Sets Sequences Strings Integers Matrices Mathematical Structures

Discrete StructuresPart 1 Sets Sequences Matrices

Dr Jorge A Peacuterezjaperez[at]rugnl

University of Groningen

November 12 2014

Jorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 161


Overview of Part 1

1 Sets and Subsets

2 Operations on Sets

3 Sequences

4 Strings

5 Integers

6 Matrices

7 Mathematical Structures



Sets Collections of Distinct Objects (1)

bull V = horse cylinder teapot mars formulabull p isin V p is an element of the set Vbull p q isin V is shorthand notation for p isin V q isin Vbull p 6isin V p is not an element of the set V so p 6isin V lArrrArr not(p isin V )1

bull Example horse isin V but rabbit 6isin V

1 lArrrArr if and only ifJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 461



bull A set has no ordering1 2 3 is the same set as 3 1 2bull Each member of a set occurs only oncea a b is the same set as a bbull A structure in which elements may occur more than once is called a bag

or multisetbull Two sets A and B are equal if they contain the same elements

Formally A = B holds whenever x isin A lArrrArr x isin Bbull The empty set contains no members by definition Notation

or empty



Defining Sets Using Predicates

bull Let P (x) be a predicate about the object xFor instance ldquox is an integer between 10 and 20 rdquobull The set of all x that satisfy the predicate P (x) is written as

x | P (x) or x P (x)

bull We may explicitly specify the domain from which x is takenFor instance given the integers Z we could have

x isin Z | P (x)



Sets of Numbers

Z = x | x is an integer= minus3minus2minus1 0 1 2 3

N = x | x is a positive integer or zero= 0 1 2 3

Z+ = N+ = x | x is a positive integer= 1 2 3

Q = x | x is a rational number

=pq| p q isin Z q 6= 0

=pq| p isin Z q isin N+

R = x | x is a real numberC = x | x is a complex number

= x+ iy | x y isin RJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 761


Subsets

bull Given two sets S and T we say that S is a subset of T (notationS sube T ) iff each element of S is also an element of T bull More precisely

S sube T lArrrArr forallx (x isin S =rArr x isin T )

Above forall denotes the universal quantifier read ldquofor allrdquobull For every set S we have S sube S and empty sube Sbull We say that S is a proper subset of T (notation S sub T ) if bothS sube T and S 6= T bull Example Z+ sub N sub Z sub Q sub R sub C



Equality of Sets

bull A = B holds whenever x isin A lArrrArr x isin B

bull In other words

A = B lArrrArr A sube B and B sube A



Sets of Sets

bull Elements of a set may be sets themselves For instance

a b a a b

bull Therefore the set a is not the same as abull Important Set membership (isin) concerns an element and a set subset

inclusion (sube) concerns two setsFor instance if S = a b c then a isin S and a sube Sbull Question Is the set empty



Russellrsquos paradox2

bull Consider the set R of all sets that are not members of themselves

R = V | V isin V

bull Question R isin R Answer yes and nobull To avoid this type of problems we consider a context which is given by a

well-defined universal set U of objectsAny set within this context is a subset of U

2Discovered by Bertrand Russell in 1901Jorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 1161


Intervals of R

50 3minus4 minus3 minus2 minus1 1 2 4

A subset of R defined by two real numbers a and b with the property thatany real between a and b is included in the setbull [a b] = x isin R | a 6 x 6 bbull (a b) = x isin R | a lt x lt bbull [a b) = x isin R | a 6 x lt bbull (a b] = x isin R | a lt x 6 bbull (minusinfin a] = x isin R | x 6 abull (a a] = [a a) = (a a) = emptybull a le b rArr [a b) = (a b] = (a b) = emptybull Jorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 1261


Power Set

bull The power set P(S) of S is

P(S) = T | T sube S

bull Example If S = a b c then

P(S) =empty a b c a b b c a c a b c



Size Cardinality

bull A finite set is a set that has a finite number of elementsbull A set that is not finite is called infinitebull Example the set 123 is finite the interval [1 3] is infinitebull The cardinality (or size) of a finite set S is denoted as |S|bull Let S be a finite set The cardinality of P(S) is 2|S|

For instance if |S| = 3 then |P(S)| = 23 = 8



Power set Binary images

All possible 2times 2 arrangements with 0123 or 4 foreground pixels|P(S)| = 2|S| = 24 = 16



Set Operations (1)

Let us write U to denote the universebull Union A cupB = x | x isin A or x isin B (or ldquoorrdquo)3

bull Intersection A capB = x | x isin A and x isin B (and ldquoandrdquo)bull Difference AminusB = x | x isin A and x isin B

Alternative notation A B

3Inclusive or x isin A or x isin B or bothJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 1761


Set Operations (2)

bull Symmetric difference

AB =x | (x isin A and x isin B) or (x isin B and x isin A)

= (A B) cup (B A)

Alternative notation A∆Bbull Complement A = U A Alternative notation Ac



Algebraic Laws of Set Operations (1)

Commutativity A cupB = B cupAA capB = B capA

Associativity (A cupB) cup C = A cup (B cup C)(A capB) cap C = A cap (B cap C)

Distributivity (cup over cap) A cup (B cap C) = (A cupB) cap (A cup C)(cap over cup) A cap (B cup C) = (A capB) cup (A cap C)




Idempotence A cupA = AA capA = A

Identity A cup empty = AA cap U = A

Complement A cupA = U

A capA = empty

Double Complement (A) = A

Relation universe - Empty set U = emptyempty = U

De Morgan Laws (A cupB) = A capB(A capB) = A cupB



Annotated Linear Proof (ALP)

Proposition A cup (B cup C) = (A cupB) cup CProof 4

x isin A cup (B cup C)

equiv definition cup (x isin A) or (x isin B cup C)

equiv definition cup x isin A or (x isin B or x isin C)

equiv logic (x isin A or x isin B) or x isin C

equiv definition cupx isin A cupB or x isin C

equiv definition cupx isin (A cupB) cup C4Kolman et al write equiv instead of lArrrArr



Sequences

bull A sequence is a list (or array) of objectsbull Unlike sets elements of a list are ordered there is a first element the

second element etcbull More formally a sequence is an indexed array sn n = 1 2 3

We call n the indexbull Example s = a b c d e is a sequence with s1 = a s2 = b etcbull Example an infinite sequence defined by sn = 2 middot snminus1 + 7 with s0=3

and n isin NWe have that s1 = 13 s2 = 33 s3 = 73



Strings

bull A string is a sequence of letters (or symbols)bull Example William (finite sequence)bull Example b l a b l a b l a b l a (infinite sequence)bull We also write blablablabla bull The alphabet of a string is the set of all letters of the string

For example the alphabet of the above string is b l a



Characteristic Function

bull Let A be a subset of the universe U We say that fA U rarr 0 1 is the characteristic function (orindicator function) of A It is defined (for all x isin U) as

fA(x) =

1 if x isin A0 if x isin A

bull Alternative notation χA(x)



Computer Representation of Sets

bull Each finite universe U can be represented by a finite sequencebull For example the set U = a c s q t corresponds to the sequenceα = a c s q tbull Every subset A sube U can be represented by its characteristic function fAbull We can make an array A of length n = |U | where A[k] = 1 iffA(α(k)) = 1 and A[k] = 0 if fA(α(k)) = 0

Representation of U 1 1 1 1 1

Representation of A = c q 0 1 0 1 0



Countable and Uncountable Sets

bull A set V is called countable (or denumerable) if V can be represented bya sequencebull Each finite set is countablebull Infinite sets that are not countable are called uncountablebull Examples sets N Z an Q are countable

The set R is not countable - the proof uses Cantorrsquos diagonal-argument5

5A proof method devised in 1891 by Georg CantorJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 2761


Languages

bull Alphabet non-empty finite set of letters (symbols) Σbull WordString a finite string of letters from alphabet Σ

Example a1 a2 a3 an where ak isin Σ is a word of length nWe usually write it as a1a2a3 anbull Σlowast denotes the set of all words that can be produced with Σ

bull Every subset of Σlowast is a languagebull The empty string written Λ isin Σlowast has length 06

Examples

bull Let Σ = abull Some words in Σlowast a aa aaabull The set of all possible words is Σlowast = Λ a aa aaa aaaa

6In most books the empty string is denoted by the symbol εJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 2961


String Concatenation

bull Consider the two strings

w1 = itisrainingcatsw2 = anddogs

bull We can append w2 to w1 so as to obtain

w1 middot w2 = itisrainingcatsanddogs

bull This operation is called the concatenation of w1 and w2We may sometimes write w1w2 instead of w1 middot w2

bull Note wΛ = Λw = w



The DNA Alphabet

Σ = A TCG



Words in the DNA alphabet

Let Σ = A TCG Let us have a look at some words from Σlowastbull Λ AAAAAAAAAA We denote this set as Alowast

Similarly for T lowast Clowast and Glowastbull ATAG This set is denoted by A(T orG)

Alternative notation A(T |G)bull ATATCGATCGCGATCGCGCGATCGCGCGCG

This set is denoted by AT (CG)lowastbull ATCGATCGCGATCGCGCGATCGCGCGCG

This set is denoted by AT (CG)+7

bull ATCTATGCTGATGGCTGGATGGGCTGGG This set is denoted by (A or C)TGlowast

7For completeness sake This notation is not used in Kolman et alJorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 3261


Regular Expressions

Let Σ be an alphabet A regular expression (regexp) over Σ is definedrecursively in terms of members of the alphabet and the symbols

( ) or lowast Λ

according to the following definitionRE1 The symbol Λ is a regular expressionRE2 Each x isin Σ is a regular expressionRE3 If α and β are regular expressions then the expression α middot β is also

regularRE4 If α and β are regular expressions then the expression α or β is also

regularRE5 If α is a regular expression then (α)lowast is also a regular expression



Regular Sets

Associated to each regexp over A there is a subset of Alowastbull Λ corresponds to the set Λbull x isin Σ corresponds to the set xbull If α and β are regexps that correspond to sets MN sube Σlowast thenbull α middot β corresponds to the set M middotN = s middot t | s isinM and t isin Nbull (α or β) corresponds to the set M cupN

bull If α is a regexp that corresponds to set M sube Σlowastthen (α)lowast corresponds to set Mlowast



DivisionRemainder Property of Natural Numbers

TheoremLet mn be integers with n gt 0There are unique integers q and r such that

n = m middot q + r and 0 6 r lt m

While q is called quotient is r called remainderWe can find them using the operators div and modbull q = n div qbull r = n mod q



Factors and Products (1)

We consider the set of integers Zbull We write a | b if b = k middot a for some integer k

We say that a is a divisor of b or that b is a multiple of abull Example 7|98 for 98 = 7 middot 14bull ldquo|rdquo is transitive (a | b) and (b | c) =rArr a | c




DefinitionA positive natural number n is a prime number if it has exactly twounique divisors 1 and n

Thus by definition 1 is not a prime Still 2 3 5 and 7 are prime




Theorem (prime factorization)

Any positive integer n gt 1 can be uniquely written as a product of primenumbers

Consider the Hasse diagram of 60

60 = 2 middot 2 middot 3 middot 5 = 22 middot 3 middot 5168 = 2 middot 2 middot 2 middot 3 middot 7 = 23 middot 3 middot 7192 = 2 middot 2 middot 2 middot 2 middot 2 middot 2 middot 3 = 26 middot 3



There are infinitely many prime numbers

A Proof by Contradiction1 Suppose there are only finitely many primes say p1 p2 pk in

increasing order2 Let n be defined as (p1 times p2 times times pk) + 13 By assumption n not a prime n is greater than the greatest prime pk

So n must be divisible by at least two primes4 The number nminus 1 is divisible by p1 p2 pk

Two numbers that differ by 1 can not have the same prime divisorSo n is not divisible by any of the primes p1 p2 pk

5 Therefore the prime factors of n are not in the list p1 p2 pkThis contradicts the assumption that p1 p2 pk is the completelist of prime numbers

6 We conclude that there are infinitely many primes



GCD

bull If a b and k are positive integers and k | a and k | b then k is acommon divisor of a and b If d is the largest such k then d is thegreatest common divisor (or GCD) We write gcd(a b) = dbull Example 3 and 2 are common divisors of 18 and 24

But gcd(18 24) = 6bull m and n are called relatively prime (or coprime) if gcd(mn) = 1

Example 17 and 24 are coprimebull A procedure for computing the CGD of two integers is the Euclideanalgorithm Given a = k middot b+ r1 it exploits the property

gcd(a b) = gcd(b r1) = gcd(r1 r2) = middot middot middot = gcd(rnminus1 rn)

In the example above (a = 24 b = 18) gcd(24 18) = gcd(18 6) = 6



LCM

If a | k and b | k then k is a common multiple of a and bThe smallest such k is the least common multiple (or LCM)We write lcm(a b) = k Example lcm(18 24) = 72

For any two positive natural numbers m and n we havebull gcd(mn) middot lcm(mn) = m middot nbull any common divisor of m and n is a divisor of gcd(mn)

bull any common multiple of m and n is a multiple of lcm(mn)



GCD and LCM

bull Greatest common divisor (GCD)

gcd(168 192) = gcd(23 middot 3 middot 7 26 middot 3)

= 2min(36) middot 3min(11) middot 7min(10) = 23 middot 31 middot 70 = 24

bull Least common multiple (LCM)

lcm(168 192) = lcm(23 middot 3 middot 7 26 middot 3)

= 2max(36) middot 3max(11) middot 7max(10) = 26 middot 31 middot 71 = 1344



Representation of Integers

bull Let b gt 1 be some integer which we call the baseEvery positive integer n can be written in the unique form

n = dk bk + dkminus1 b

kminus1 + d1 b1 + d0

where 0 le di lt b i = 0 1 k and dk 6= 0bull The sequence dkdkminus1 d1d0 is called the base b expansion of nbull b = 10 decimal expansionbull b = 2 binary expansionbull b = 8 octal expansionbull b = 16 hexadecimal expansion



Definition of a Matrix

bull A matrix is a rectangular array of numbers arranged in (horizontal) rowsand (vertical) columns For instance

A = [aij ] =

a11 a12 middot middot middot a1Na21 a22 middot middot middot a2N

aM1 aM2 middot middot middot aMN

bull M and N are called the dimensions of the matrix

We say that the matrix is M by N written as M timesN bull If M = N we say that the matrix is a square matrix of order N



Transpose of a Matrix

bull The transpose of a matrix A is the matrix obtained by interchangingthe rows and columns of A Example Let

A =



The transpose of A denoted AT is defined as

AT =

a11 a21 middot middot middot aM1


a1N a2N middot middot middot aMN

bull A matrix is called symmetric if A = AT Jorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 4761


Special Matrices

Square matrices (n rows n columns)

Diagonal matrix

Dn =

a11 0 middot middot middot 00 a22 middot middot middot 0

0 0 middot middot middot ann

Identity matrix

In =

1 0 middot middot middot 00 1 middot middot middot 0

0 0 middot middot middot 1

Zero matrix

Zn =





Addition of Matrices

bull Consider two matrices A and B with the same dimensionbull The sum of A and B is computed by adding the corresponding

elements For instance when A and B have dimensions 3times 2

A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32



Product of Matrices (1)

bull The product of an M times P matrix A and a P timesN matrix B is theM timesN matrix C = [cij ] denoted ABbull To compute cij take row i of A and column j of B multiply the

corresponding elements and sum all these productsbull Example Consider M = 2 P = 3 and N = 2 We have

A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32

The product AB is the 2times 2 matrix defined as follows(

c11 = a11b11 + a12b21 + a13b31 c12 = a11b12 + a12b22 + a13b32c21 = a21b11 + a22b21 + a23b31 c22 = a21b12 + a22b22 + a23b32

)Jorge A Peacuterez Discrete Structures (2014-2015) ndash Part 1 5061



bull A concrete example

A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13

)In fact it is easy to verify that

c11 = (2 middot 7) + (3 middot minus6) + (minus4 middot 9) = minus40

c12 = (2 middot 1) + (3 middot 6) + (minus4 middot 0) = 20

c21 = (1 middot 7) + (2 middot minus6) + (5 middot 9) = 40

c22 = (1 middot 1) + (2 middot 6) + (5 middot 0) = 13

bull The matrix product is associative ie A(BC) = (AB)C

bull In general the matrix product is not commutative ie AB 6= BA



Boolean Matrices

Figure Mathematician George Boole (1815-1864)

bull In a Boolean matrix elements are either 1 (ldquotruerdquo) or 0 (ldquofalserdquo)bull Let A = [aij ] and B = [bij ] be Boolean matrices We define

bull Join A orB = C = [cij ] where cij =

1 if aij = 1 or bij = 1

0 if aij = bij = 0

bull Meet A andB = C = [cij ] where cij =

1 if aij = bij = 1




Boolean Product of Matrices (1)

bull The product of Boolean matrices is similar to that of ordinary matricesbull Given A and B we denote their product as AB = C = [cij ]bull To compute element cij we take row i of A and column j of B multiply

the corresponding elements and take the maximum of all these productsmdash which can be either zero or onebull A particular case

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=

(max(a11b11 a12b21 a13b31) max(a11b12 a12b22 a13b32)max(a21b11 a22b21 a23b31) max(a21b12 a22b22 a23b32)




bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=

(M(a11b11 a12b21 a13b31) M(a11b12 a12b22 a13b32)M(a21b11 a22b21 a23b31) M(a21b12 a22b22 a23b32)

)

where M(n) =

0 if n = 0

1 if n gt 0



Mathematical Structures

Definition (Mathematical Structure (MS))

A mathematical structure is a collection of objects with operations definedon them (and their associated properties)

bull Notation(collection operation1 operation2 operationn)

bull Examples

(R+times) the real numbers with the operationsaddition and multiplication

(P (U)cupcap middot) the power set of U with the operationsunion intersection and complement



Closures

bull A MS (VO1 On) is closed with respect to an operation Oi if theresult of applying Oi yields again an element of V bull Example Strings are closed with respect to concatenationbull Type of operations Obull Unary O has one argument

Example Complement of a set O(A) = Abull Binary O has two arguments

Example Sum of two integers O(x y) = x+ ybull Ternary O has 3 argumentsbull



Prefix Infix Postfix Notation

bull The postfix notation is typical of unary operationsFor instance we write AT for matrix transposition mdashas opposed to theprefix notation T (A)bull The infix notation is commonly used to denote binary operations

For instance we A cupB for set union mdashas opposed to the prefixnotation cup(AB)



Properties of Operations (1)

bull A unary operation in a MS (V ) is called idempotent if

(x) = x forallx isin V

bull A binary operation in a MS (V ) is calledbull associative if for all x y z isin V

(x y) z = x (y z)

bull commutative if for all x y isin V

x y = yx

bull An element e is called an identity element of if for all x isin V

x e = ex = x




bull Let (V O ) be a MS with binary operations and OWe say that distributes over O if for all x y z isin V

x (yO z) = (x y)O (x z)

bull Let (V O ) be a MS with a unary operation and binary operations and O The operations satisfy De Morganrsquos laws if for x y isin V

(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Overview of Part 1

1 Sets and Subsets

2 Operations on Sets

3 Sequences

4 Strings

5 Integers

6 Matrices

7 Mathematical Structures












or empty







x isin Z | P (x)



Sets of Numbers










Subsets






Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices












or empty







x isin Z | P (x)



Sets of Numbers










Subsets






Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices







x isin Z | P (x)



Sets of Numbers










Subsets






Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Sets of Numbers










Subsets






Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Subsets






Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Equality of Sets


bull In other words




Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Sets of Sets


a b a a b







R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices





R = V | V isin V





Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Intervals of R




Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Power Set


P(S) = T | T sube S





Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Size Cardinality









Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices







Set Operations (1)






Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Set Operations (2)



= (A B) cup (B A)














A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices













A capA = empty
















Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices













Sequences







Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Strings







fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices





fA(x) =
















Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices














Languages




Examples













The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices












The DNA Alphabet

Σ = A TCG












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices












Regular Expressions


( ) or lowast Λ






Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Regular Sets





































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



































GCD








LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



LCM






GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



GCD and LCM


















A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices












A = [aij ] =









A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices





A =




AT =






Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Special Matrices


Diagonal matrix

Dn =



Identity matrix

In =



Zero matrix

Zn =








A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices






A+B =

a11 a12a21 a22a31 a32

+

b11 b12b21 b22b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22a31 + b31 a32 + b32






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices






A =

(a11 a12 a13a21 a22 a23

)B =

b11 b12b21 b22b31 b32







A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices





A =

(2 3 minus41 2 5

)B =

7 1minus6 69 0

AB = [cij ] =

(minus40 2040 13










Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Boolean Matrices





0 if aij = bij = 0


1 if aij = bij = 1







AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices






AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=





bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices




bull Another view

AB =

(a11 a12 a13a21 a22 a23

)

b11 b12b21 b22b31 b32

=


)

where M(n) =

0 if n = 0

1 if n gt 0







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices







bull Examples





Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



Closures















(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices












(x y) z = x (y z)


x y = yx


x e = ex = x





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices





x (yO z) = (x y)O (x z)


(x y) = xO y

(xO y) = x y



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices



End of Part 1


Sets and Subsets

Operations on Sets

Sequences

Strings

Integers

Matrices


sets, sequences, matrix

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