Lecture3: Signed Numbers and Operation on Series of Numbers

SIGNED NUMBERS & OPERATION IN SERIES OF NUMBERS

SIGNED NUMBERS

·        These are numbers preceded by a plus (+) or minus ( – ) sign including zero.

·        Another term for integers.

FOUR FUNDAMENTAL OPERATIONS WITH SIGNED NUMBERS

ADDITION

R1. In adding signed numbers having common sign, add the numbers and affix the common sign.

e.g.:

1.)  2 + 4 + 6 = 12            2.)    (-12) + (-6) + (-8) = -28

R2. In adding signed numbers having unlike sign, subtract the two numbers from the largest to the smallest and affix the sign of the larger number.

e.g.:

1.)  5 + (-3) = 2                2.)    -5 + 3 = -2                   3.)    12 + (-15) = -3

SUBTRACTION

R1. In subtracting signed numbers, change the sign of the subtrahend and proceed as in addition.

e.g.:

1.)  115 – 3 = 112            2.)    25 – 75 = -50             3.)    -2 – (-3) = 1

4.)     (-15) – (-2) = -13   5.)    -2 – 13 = -15              6.)   14 – (-2) = 16

MULTIPLICATION

R1. In multiplying signed numbers with like sign, multiply the numbers and always affix the positive sign to the result.

e.g.:

1.)  25 * 4 = 100             2.)   -8 * -9 = 72

R2. In multiplying signed numbers with unlike sign, multiply the numbers and always affix the negative sign to the result.

e.g.:

1.)  15 * (-3) = -45          2.)   -6 * 4 = -24

DIVISION

R1. In dividing signed numbers with like sign, divide the numbers and always affix the positive sign to the result.

e.g.:

1.)  18 / 2 = 9                 2.)   -25 / -5 = 72

R2. In dividing signed numbers with unlike sign, divide the numbers and always affix the negative sign to the result.

e.g.:

1.)  15 / (-3) = -5             2.)   -81 / 3 = -27

OPERATION IN SERIES OF NUMBERS

·        In performing operations in series of numbers in mathematics the GEMDAS rule must be applied.

·        GEMDAS is the order of operation in mathematics which stands for Grouping symbol, Exponents, Multiplication, Division, Addition and Subtraction.

·        There are various grouping symbols in mathematics like ( ) parenthesis, [ ] brackets, { } curly braces and         vinculum.

·        In applying the GEMDAS rule, the inner grouping symbol must be simplified first.

e.g.:

1.)  2 + 2 / 2

= 2 + 1

= 3

2.)  16 / 8 + 4 * 2 – 2

=16 / 8 + 8 – 2

=2 + 8 – 2

=10 – 2

=8

3.)  – [2 + 4(3) – 2] + 3 [13(2 – 5 + 12) – 8]

= – [2 + 12 – 2] + 3 [13(9) – 8]

= – [2 + 12 – 2] + 3 [117 – 8]

= – [12] + 3 [109]

= – 12 + 327

= 315

4.)  6 + {2 – [5 – 6 + 3(2 – 4 + 5)] – 2 + [18 – 8 / 2 + 6(4 – 8)]}

= 6 + {2 – [5 – 6 + 3(3)] – 2 + [18 – 8 / 2 + 6(–4)]}

= 6 + {2 – [5 – 6 + 9] – 2 + [18 – 8 / 2 –24]}

= 6 + {2 – [5 – 6 + 9] – 2 + [18 – 4 –24]}

= 6 + {2 – [8] – 2 + [–10]}

= 6 + {2 – 8 – 2 – 10}

= 6 + {– 18}

= 6 – 18

= – 12

Lecture2: Set Theory

SET THEORY

SET, Defined:

·        is a collection of well-defined distinct objects.

e.g.: computer set = {mouse, printer, keyboard, CPU, monitor}

NAMING A SET

·        A set can be named using any capital letter in the English alphabet except from U.

·        Each object in a set is called “elements”.

            e.g.: C = { mouse, printer, keyboard, CPU, monitor}

                     V = {a, e, i, o, u}

                     A = {1, 2, 3, 4, 5}

WAYS IN DESCRIBING A SET

1.)  Tabular/Roster method – each elements are listed, separated by commas and enclosed in a curly braces.

e.g.: C = { mouse, printer, keyboard, CPU, monitor}

         V = {a, e, i, o, u}

         A = {1, 2, 3, 4, 5}

2.)  Rule method – it makes use of the description {x|x is an element of…}, which read as “x such that x is an element of…”

e.g.: C = {x|x is an element of computer hardware}

         V = {x|x is an element of vowels in the English alphabet}

         A = {x|x is an element of the first five positive integers}

           

KINDS OF SET

1.)  Equal set – are sets containing exactly the same elements.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {2, 4, 1, 3, 5}

            Therefore, A = B

2.)  Equivalent set – are sets with the same number of elements.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {a, e, i, o, u}

            Therefore, A ~ B

3.)  Finite set – contains countable number of elements.

e.g.:    A = {a, e, i, o, u}

            B = { x|x is an element of months in a year}

4.)  Infinite set – contains uncountable number of elements.

e.g.:    Z = {… , -3, -2, -1, 0, 1, 2, 3, …}

            B = {x|x is an element of grains of rice}

5.)  Joint set – are sets with elements in common.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {2, 4, 6, 8, 10}

            A and B are joint set for it contains element in common, 2 and 4.

6.)  Disjoint set – are sets with no elements in common.

e.g.:    A = { x|x is an element of even numbers}

            B = { x|x is an element of odd numbers}

            A and B are disjoint set for it contains no element in common.

7.)  Universal set – is the totality of elements under consideration denoted as U.

8.)  Subset – is a part of a given set.

e.g.:    A = {1, 2, 3}

            {1} Ì A                      {2} Ì A                      {3} Ì A

            {1, 2} Ì A                  {1, 3} Ì A                  {2, 3} Ì A

            {1, 2, 3} Ì A

9.)  Null  Æ or Empty {} set – a set containing no elements.

VENN DIAGRAM

·        is a picture representation of sets, developed by John Venn in 1880, which makes use of rectangle as the universal set and circles as its subsets.

OPERATIONS ON SET

1.)  Union (È) – is the combination of elements in two or more sets without repetition.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

            AÈB = {1, 2, 3, 4, 5, 6, 8, 10}

*some reference uses (Ú) “or” symbol

to denote Union.

2.)  Intersection (Ç) – are the element common in the given sets.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

            AÇB = {2, 4}

*some reference uses (Ù) “and” symbol

to denote Intersection.

3.)  Complement (‘) – these are elements found in the universal set but not on the given set.

e.g.:    U = {0, 1, 2, 3, 4, 5}

            A = {2, 3, 4}

            A’ = {0, 1, 5}

  

4.)  Difference ( – ) – these are elements found in a given set but not on the other set.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

          A – B = {1, 3, 5}

          B – A = {6, 8, 10}

GENERAL RULES OF SET

1.)  Union

A È U = U                 A È Æ = A                A È A = A

2.)  Intersection

A Ç U = A                 A Ç Æ = Æ               A Ç A = A

3.)  Complement

A’ È U = U                A’ È A = U                A’ Ç A = Æ

4.)  Difference

A – B = B’ Ç A          B – A = A’ Ç B

 PROPERTIES/ LAWS OF SET

1.)  Commutative Law

A È B = B È A                                              A Ç B = B Ç A

2.)  Associative Law

A È (B È C) = (A È B) È C                       A Ç (B Ç C) = (A Ç B) Ç C

3.)  Distributive Law

A È (B Ç C) = (A È B) Ç (A È C)                        A Ç (B È C) = (A Ç B) È (A Ç C)           

4.)  Identity Law

A È Æ = A                                                    A Ç U = A

5.)  Inverse Law

A’ È A = U                                                    A’ Ç A = Æ

APPLICATION OF SET THEORY

Sample Problem:

In an excursion at Enchanted Kingdom, 80 students brought Sandwich, Drinks and Chips as follows:

50 students brought Sandwich (S)

30 students brought Drinks (D)

30 students brought Chips (C)

18 students brought Chips and Drinks

15 students brought Sandwich and Chips

8 students brought Sandwich and Drinks

6

5 students brought all the said stuffs.

  

Construct a Venn Diagram and answer the following questions:

How many students brought:

1.)  Sandwich only? Drinks only? Chips only?       32;      9;       2

2.)  Nothing?                                                                   6

3.)  Sandwich and chips but not drinks?                          10      

4.)  Chips or drinks but not sandwich?                            24

5.)  Exactly one of the stuffs?                                         43

6.)  At most two of the stuffs?                                        75

7.)  At least two of the stuffs?                                         31

Lecture1: Overview

INTRODUCTION

ALGEBRA, Defined:

·        Came from two Arabic word: Al – means “the” and Jabara – means “reunion”

·        Literally, it is called as “the subject of reunion”, because of combining numbers and letters for computation.

·        A branch of mathematics which deals with numbers and letters that is use for computations.

IMPORTANCE OF ALGEBRA

·        This branch of mathematics have various importance in its application. But its main significance is, Algebra, serves as a preparatory subject for better understanding of higher mathematics.

NUMBERS vs. NUMERALS

·        Numbers – are abstract ideas of certain quantity.

·        Numerals – are symbols or representation of numbers.

REAL NUMBER SYSTEM

     ·        Zero = {0}

·        Positive/ Natural/ Counting numbers = {1, 2, 3, 4, …}

·        Whole numbers/ non-negative integers = {0, 1, 2, 3, 4, …}

·        Negative Integers = {-1, -2, -3, -4, …}

·        Integers = {… , -3, -2, -1, 0, 1, 2, 3, …}

·        Fractions = {…, ¼, 1/3, ½, …}

·        Decimals = {…, 0.25, 0.333, 0.5, …}

·        Rational numbers = {D, F, Z}

·        Irrational numbers = {sqrt 2, π, …}

·        Real numbers = {Q, Q’}

Additional definition:

·        Even numbers = {...,-4, -2, 0, 2, 4, …}

·        Odd numbers = {…, -5, -3, -1, 1, 3, 5, …}

·        Prime numbers = {1, 2, 3, 5, 7, 11, …}

·        Composite numbers = {4, 6, 8, 9, 10, …}

PROPERTIES OF EQUALITY

1.)  Reflexive: a = a                                             e.g.: 2 = 2

2.)  Symmetric: if a = b, then b = a                      e.g.: if x = 5, then 5 = x

3.)  Transitive: if a = b and b = c, then a = c        e.g.: if x = y and y = 3, then x = 3

4.)  Addition Property of Equality (APE)

If a = b, then a + c = b + c                           e.g.: if x = 2, then x + 4 = 2 + 4

5.)  Multiplication Property of Equality (MPE)

If a = b, then ac = bc                                    e.g.: if x = 5, then x(3) = 5(3)

6.)  Substitution

PROPERTIES OF REAL NUMBER

1.)  Closure – a real number added, subtracted, multiplied or divided to another real number, the result is always a real number.

e.g.:    Addition: 3 + 15 = 18

            Subtraction: 4 – 19 = -15

            Multiplication: 6 * 2 = 12

            Division: 27/3 = 9

2.)  Commutative – two numbers can be added/ multiplied in any order.

            Addition: a + b = b + a                         e.g.: 5 + 7 = 7 + 5

            Multiplication: ab = ba                          e.g.: 6(-4) = (-4)6

3.)  Associative – when three numbers are added/ multiplied, it makes no difference on either which two are added/ multiplied first.

Addition: (a + b) + c =a + (b + c)   e.g.: (2 + 5) + 8 =2 + (5 + 8)

Multiplication: (ab)c =a(bc)             e.g.: (5*2)*3 =5*(2*3)

4.)  Distributive – Multiplication distributes over addition.

a(b + c) = ab + ac                       e.g.: 4(3 + 6) = 4*3 + 4*6

5.)  Identity – the sum of a number and zero or the product of a number and one is equal to the number itself.

Addition: a + 0 = a                       e.g.: 3 + 0 = 3

Multiplication: a * 1 = a                e.g.: 8 * 1 = 8

6.)  Inverse – the sum of a number and its negative inverse is equal to 0 or the product of a number and its reciprocal is equal to 1.

Addition: a + (-a) = 0                   e.g.: 2 + (-2) = 0

Multiplication: a*(1/a) = 1             e.g.: 6*(1/6) = 1

7.)  Cancellation

ABSOLUTE VALUE

·        It is the number itself regardless of its sign. (||)

e.g.:

1.)   |-13| = 13                               2.)   |5| = 5

3.)   |-|-15|| = 15                         4. )   - |5|= -5