Mathematics I - Lanka Education and Research …scholar.sjp.ac.lk/.../files/lanel/files/lectures_1_and_2.pdfIntroduction Mathematics and mathematical science is a study which analyzes

Mathematics I

Dr. G.H.Jayantha Lanel

Lectures 1 and 2

Dr. G.H.Jayantha Lanel (USJP) Mathematics I Lectures 1 and 2 1 / 54

Outline

Outline


Introduction

Outline1 Introduction2 Introduction to Mathematical logic

Logical operatorsTautologies and contradictionsLogically equivalenceCompound propositions with examples

3 Revisiting number systemsDivisibility in Z

4 Counting principlesFactorialsBasic idea of permutations and combinations using examples

5 Set theorySubsetEquality of two setsSet operationsProduct of setsConcept of the power setAlgebra of setsDe Morgan’s laws


Introduction

Mathematics has played and continues to play a critical role inexpanding fields of science and technology.

It is mainly because of the requirement that research needs to beable to quantify and accurately evaluate the results of changes ina field of endeavour.


Introduction

Mathematics has played and continues to play a critical role inexpanding fields of science and technology.

It is mainly because of the requirement that research needs to beable to quantify and accurately evaluate the results of changes ina field of endeavour.


Introduction

To develop new treatments for medical conditions knowing theamount of medication administered is critical.

Mathematical formulas and models may be applied to test thestructural and functional soundness of the design before it is massproduced.

Computer-aided design is increasingly important in this type ofuse and it is all based on application of mathematics.


Introduction





Introduction





Introduction

When introducing mathematical theorems, there is no such wordbeyond expectations.

Every possibility is identified and a single theorem is constructed.


Introduction

When introducing mathematical theorems, there is no such wordbeyond expectations.

Every possibility is identified and a single theorem is constructed.


Introduction

Mathematics is said to be a useful study for acquiring logicalthinking.

Mathematical results can be useful in society.

Mathematics is a study that nurtures logical thinking for makingappropriate predictions for the future in today’s complicated world.


Introduction





Introduction





Introduction

Mathematics and mathematical science is a study which analyzesrisks associated with nuclear power, hydrogen infrastructure andspace systems, and provides useful problem solving technology.

Academic needs of mathematics and mathematical science haveintensified rapidly.

Problems posed in each field provide feed back for mathematics,leading to expectations for the development of new academicfields within the framework of mathematics.


Introduction





Introduction





Introduction to Mathematical logic







Introduction to Mathematical logic Logical operators

Logical operators

Logical operators are symbols that help to write out a logical sentenceusing expressions instead of words.

Five logical operators are:

NOTANDOREQVNEQV



Logical operators

Logical operators are symbols that help to write out a logical sentenceusing expressions instead of words.

Five logical operators are:

NOTANDOREQVNEQV




Introduction to Mathematical logic Tautologies and contradictions

Tautology

A tautology is a formula which is always true i.e. it is true for everyassignment of truth values to its simple components.

Contradiction

The opposite of a tautology is a contradiction, a formula which isalways false. In other words, a contradiction is false for everyassignment of truth values to its simple components.



Tautology

A tautology is a formula which is always true i.e. it is true for everyassignment of truth values to its simple components.

Contradiction

The opposite of a tautology is a contradiction, a formula which isalways false. In other words, a contradiction is false for everyassignment of truth values to its simple components.




Introduction to Mathematical logic Logically equivalence

Logically equivalence

The formulas A and B are said to be logically equivalent when A istrue if and only if B is true (i.e. A⇒ B and B⇒ A)

Example:

1 A real number is positive if and only if it is greater than zero.

2 He is a bachelor if and only if he is an unmarried man.





Example:







Example:




Introduction to Mathematical logic Compound propositions with examples

Proposition

A declarative sentence that is either true or false but not both.

Example:Pigs can fly.Today is Thursday.

Non-example:Do your homework.When is my birthday.



Proposition






Proposition






Proposition






Proposition






Proposition






Proposition






Compound proposition

One that can be broken down into more primitive propositions.

Example:If it is a sunny outside then I can walk to work; otherwise I drive, and ifit is raining then I can carry umbrella.

This consists of several primitive propositions:

p=It is a sunny outside, q=I walk to work, r=I drive, s= It is raining andt=I carry my umbrella.

Connectives: if ... then, otherwise, and

If p then q; otherwise r and if s then t .If p then q and (if not p then (r and (if s then t))).p implies q and ((not p) implies (r and (s implies t))).
























































Revisiting number systems








Natural numbers - N

The natural (or counting) numbers are 1, 2, 3, 4, 5 etc.

There are infinitely many natural numbers.

The set of natural numbers, {1, 2, 3, 4, 5,. . . } is sometimes written Nfor short.



Natural numbers - N






Natural numbers - N






The whole numbers are the natural numbers together with 0.

The sum of any two natural numbers is also a natural number and theproduct of any two natural numbers is a natural number.

This is not true always for subtraction and division.













Integers - Z

The integers are the set of real numbers consisting of the naturalnumbers, their additive inverses and zero.

The set of integers,{. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5,. . . } issometimes written Z for short.

The sum, product, and difference of any two integers is also an integer.But this is not true for division.



Integers - Z






Integers - Z






Rational numbers - Q

The rational numbers are those numbers which can be expressed asa ratio between two integers.

All the integers are included in the rational numbers, since any integerZ can be written as the ratio Z/1.



Rational numbers - Q

The rational numbers are those numbers which can be expressed asa ratio between two integers.

All the integers are included in the rational numbers, since any integerZ can be written as the ratio Z/1.



All decimals which terminate are rational numbers (since 8.27 can bewritten as 827/100).

Decimals which have a repeating pattern after some point are alsorationals: for example,

0.0833333.... =1

12

The set of rational numbers is closed under all four basic operations.

An irrational number is a number that cannot be written as a ratio (orfraction)





0.0833333.... =1

12







0.0833333.... =1

12







0.0833333.... =1

12







0.0833333.... =1

12





Real numbers - R

The real numbers is the set of numbers containing all of the rationalnumbers and all of the irrational numbers.



Some applications.

Natural numbers are those used for counting (as ”there are sixcoins on the table”) and ordering.

Geographically, we represent sea level with integers. Obviously,below sea level is represented with negative integers. Forexample, Death Valley in California is located at eighty six metersbelow sea level. This can be represented numerically as 86 m.

Real numbers are used in car driving instruments, dashboard, fuelamount, speed, rpm and engine temperature.



Some applications.






Some applications.






Some applications.






Real numbers include the rationals, which include the integers, whichinclude the natural numbers.


Revisiting number systems Divisibility in Z

Divisibility

An integer b 6= 0 divides another integer a iff ∃k ∈ Z such that a = kb.

We also say that b is a factor (or divisor) of a.

One frequently writes b|a to indicate that b divides a.



Divisibility






Divisibility






Example:

Determine whether 150 is divisible by 2, 3, 4, 5, 6, 9 and 10.

150 is divisible by 2 since the last digit is 0.150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6),and 6 is divisible by 3.150 is not divisible by 4 since 50 is not divisible by 4.150 is divisible by 5 since the last digit is 0.150 is divisible by 6 since it is divisible by 2 and 3.150 is not divisible by 9 since the sum of the digits is 6, and 6 isnot divisible by 9.150 is divisible by 10 since the last digit is 0.

Solution: 150 is divisible by 2, 3, 5, 6, and 10.



Example:






Example:






Example:






Example:






Example:






Example:






Example:





Counting principles







Counting principles Factorials

n!, where n ∈ N

The factorial function (!) means to multiply a series of descendingnatural numbers.

Example:7! = 7× 6× 5× 4× 3× 2× 1 = 5040

What about ”0!”

It is generally agreed that 0!=1.

Factorials are used in many areas of mathematics, but particularly incombinations and permutations.



n!, where n ∈ N


Example:7! = 7× 6× 5× 4× 3× 2× 1 = 5040

What about ”0!”





n!, where n ∈ N


Example:7! = 7× 6× 5× 4× 3× 2× 1 = 5040

What about ”0!”





n!, where n ∈ N


Example:7! = 7× 6× 5× 4× 3× 2× 1 = 5040

What about ”0!”





n!, where n ∈ N


Example:7! = 7× 6× 5× 4× 3× 2× 1 = 5040

What about ”0!”





Fundamental Counting Principle

The fundamental counting principle states that if one event has mpossible outcomes and a second independent event has n possibleoutcomes then there are m × n total possible outcomes for the twoevents together.







Exercise:

Eight horses- Alabaster, Beauty, Candy, Doughty, Excellente, Friday,Greaty and Mighty run a race.

In how many ways can the first three finishes turn out?

8× 7× 6 = 336



Exercise:



8× 7× 6 = 336



Exercise:



8× 7× 6 = 336


Counting principles Basic idea of permutations and combinations using examples

Permutations

An arrangement where order is important is called a permutation.

Example:

Mary, Sandy, John and Ben are running for the office of president,secretary and treasurer. In how many ways can these offices be filled?

4× 3× 2× 1 = 24



Permutations


Example:


4× 3× 2× 1 = 24



Permutations


Example:


4× 3× 2× 1 = 24



Permutations


Example:


4× 3× 2× 1 = 24



Exercises:

1 If five digits 1, 2, 3, 4, 5 are being given and a three digit code hasto be made from it, if the repetition of digits is allowed, then howmany such codes can be formed.

As repetition is allowed, we have five options for each digit of thecode. Hence, the required number of ways code can be formed is,5×5×5=125.

2 In how many ways can the letters of the word APPLE can berearranged?

Total number of characters in APPLE = 5.Number of repeated characters = 2.Number of ways APPLE can be rearranged = 5!/2! = 60.The word APPLE can be rearranged in 60 ways.



Exercises:







Exercises:







Exercises:







Exercises:







Exercises:







Exercises:







Combinations

An arrangement where order is not important is called a combination.

Example:

Mary has four different coins in her pocket and pulls out three at onetime. How many different amounts can she get?

4× 3× 23!

= 4



Combinations


Example:


4× 3× 23!

= 4



Combinations


Example:


4× 3× 23!

= 4



Exercises:

1 How many combinations can the seven colors of the rainbow bearranged into groups of three colors each?

7× 6× 53!

= 35

2 Six friends want to play enough games of chess to be sure everyone plays everyone else. How many games will they have to play?

6× 52!

= 15

They will need to play 15 games.



Exercises:


7× 6× 53!

= 35


6× 52!

= 15




Exercises:


7× 6× 53!

= 35


6× 52!

= 15




Exercises:


7× 6× 53!

= 35


6× 52!

= 15




Determine if the situation represents a permutation or a combination:

1 In how many ways can five books be arranged on a book shelf in alibrary?Permutation

2 In how many ways can three student-council members be electedfrom five candidatesCombination

3 Seven students line up to sharpen their pencils.Permutation

4 A DJ will play three CD choices from the five requests.Combination


























































Set theory







Set theory Subset

Subset

If A and B are two sets, and every element of set A is also an elementof set B, then A is called a subset of B and we write it as A⊆B or B⊇A.

The symbol ⊂ stands for is a subset of or is contained in.

Every set is a subset of itself, i.e., A⊂A, B⊂B.

Empty set is a subset of every set.


Set theory Subset

Subset






Set theory Subset

Subset






Set theory Subset

Subset






Set theory Subset

Example:

Let A = {2,4,6} and B = {6,4,8,2}

Here A is a subset of B since, all the elements of set A are containedin set B.


Set theory Subset

Example:

Let A = {2,4,6} and B = {6,4,8,2}



Set theory Subset

Example:

Let A = {2,4,6} and B = {6,4,8,2}



Set theory Equality of two sets

Equality of two sets

Equality of sets is defined as set A is said to be equal to set B if bothsets have the same elements or members of the sets.

Example:

Let A = {x : x2 − 10x + 16 = 0} and B = {2,8}.

Then A = B.





Example:

Let A = {x : x2 − 10x + 16 = 0} and B = {2,8}.

Then A = B.





Example:

Let A = {x : x2 − 10x + 16 = 0} and B = {2,8}.

Then A = B.


Set theory Set operations

Set operations

There are 4 basic set operations: union, intersection, complement, anddifference.

Union

Set that contains all the elements in either A or B or both.A∪B = { x:x∈A or x∈B}



Set operations


Union




Set operations


Union




Intersection

Set that contains all the elements that are in both A and B.A∩B = {x :x ∈A and x∈B}



Intersection

Set that contains all the elements that are in both A and B.A∩B = {x :x ∈A and x∈B}



Complement

AC is the set that contains everything in the universal set that is not inA.AC={x : x∈ U and x /∈ A}



Complement

AC is the set that contains everything in the universal set that is not inA.AC={x : x∈ U and x /∈ A}



Difference

A-B is the set that contains all the elements that are in A but not in B.A−B={x : x∈ A and x /∈ B}



Difference

A-B is the set that contains all the elements that are in A but not in B.A−B={x : x∈ A and x /∈ B}


Set theory Product of sets

Cartesian Product

The Cartesian Product of two sets A and B is the set of all OrderedPairs (a,b) where the first element of order pairs a belongs to first set Aand second element of ordered pairs b belongs or second set B.

Note: Cartesian product of set A and B is not equal to Cartesianproduct of set B and A.

Example:

If set A={1,2} and set B={4,5}Then,A×B=[ {1,4} , {1,5} , {2,4} , {2,5} ] andB×A=[ {4,1} , {4,2} , {5,1} , {5,2} ]



Cartesian Product



Example:




Cartesian Product



Example:




Cartesian Product



Example:




Cartesian Product



Example:



Set theory Concept of the power set

Power Set

A Power set is a set of all the subsets of a set.

Example:

For the set {a,b,c}: These are subsets;{},{a},{b},{c},{a,b}, {a,c},{b,c},{a,b,c}And when we list all the subsets of S={a,b,c } we get the Power Setof {a,b,c}:

P(S)={{},{a},{b},{c},{a,b}, {a,c}, {b,c},{a,b,c}}



Power Set


Example:


P(S)={{},{a},{b},{c},{a,b}, {a,c}, {b,c},{a,b,c}}



Power Set


Example:


P(S)={{},{a},{b},{c},{a,b}, {a,c}, {b,c},{a,b,c}}



Power Set


Example:


P(S)={{},{a},{b},{c},{a,b}, {a,c}, {b,c},{a,b,c}}



Power Set


Example:


P(S)={{},{a},{b},{c},{a,b}, {a,c}, {b,c},{a,b,c}}



Exercise:

write the power set of S={1,2,,3,4}.



How many subsets???

If the original set has n members, then the Power Set will have 2n

members.



How many subsets???

If the original set has n members, then the Power Set will have 2n

members.


Set theory Algebra of sets

Algebra of sets

Algebra of sets explains the basic properties and laws of sets,i.e., the set-theoretic operations of union, intersection, andcomplementation.

It explains the relations of set equality and set inclusion.

Some of the useful properties/operations on sets are as follows:

A∪ U = UA∩ φ = φφC = UUC = φ



Algebra of sets







Algebra of sets







Algebra of sets







Algebra of sets







Algebra of sets







Algebra of sets






Set theory De Morgan’s laws

De Morgan’s Law

The complement of the union of two sets is the intersection of theircomplements and the complement of the intersection of two sets is theunion of their complements.These are called De Morgans laws.

The laws are as follows:

A∪BC = AC∩BC

A∩BC = AC∪BC



De Morgan’s Law



A∪BC = AC∩BC

A∩BC = AC∪BC



De Morgan’s Law



A∪BC = AC∩BC

A∩BC = AC∪BC



De Morgan’s Law



A∪BC = AC∩BC

A∩BC = AC∪BC



Exercise:

Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.Show that A∪ BC = AC∩BC .



Exercise:

Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.Show that A∪ BC = AC∩BC .



End!


Mathematics I - Lanka Education and Research …scholar.sjp.ac.lk/.../files/lanel/files/lectures_1_and_2.pdfIntroduction Mathematics and mathematical science is a study which analyzes

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