Logic of Mathematics (College Algebra for Computer Science)

7/31/2019 Logic of Mathematics (College Algebra for Computer Science)

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Table of Contents

PREFACE .............................................................................................. 4

SETS .................................................................................................... 5

Kinds of Sets ................................................................................... 6

Finite Sets ................................................................................... 6

Infinite Sets ................................................................................. 6

Universal Sets ............................................................................. 7

Null Sets ...................................................................................... 7

Relationship of Sets ........................................................................ 7

Equal Sets ................................................................................... 7

Equivalent Set ............................................................................. 8

Joint Sets ..................................................................................... 8

Disjoint Sets ................................................................................ 9

Subsets ........................................................................................ 9

Set Operations .............................................................................. 10

Union ........................................................................................ 10

Intersection............................................................................... 10

Difference ................................................................................. 11

Complement ............................................................................. 11

Set Product ............................................................................... 12

Venn-Euler Diagram ..................................................................... 12

REAL NUMBERS ................................................................................ 21

Kinds of Real Numbers ................................................................. 21

Rational Numbers ..................................................................... 21

Integers ..................................................................................... 22

Natural Numbers ...................................................................... 22

Whole Numbers ........................................................................ 22


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Fractions ................................................................................... 22

Decimal Numbers ..................................................................... 23

Irrational Numbers ................................................................... 23

Properties of Real Numbers ......................................................... 23

Closure ...................................................................................... 24

Commutative ............................................................................ 24

Associative ................................................................................ 24

Distributive ............................................................................... 25

Identity ..................................................................................... 25

Inverse ...................................................................................... 26

Reflexive ................................................................................... 26

Symmetric ................................................................................. 27

Transitive .................................................................................. 27

Substitution .............................................................................. 28

Addition .................................................................................... 28

Multiplication ........................................................................... 29

ALGEBRAIC EXPRESSIONS ................................................................. 37

Polynomials .................................................................................. 37

Some Terms: ............................................................................. 37

Operations on Polynomials ...................................................... 39Special Products ....................................................................... 40

Binomial Theorem .................................................................... 41

Pascal Triangle .......................................................................... 41

Factoring ................................................................................... 42

RADICALS .......................................................................................... 50

Laws of Radicals ............................................................................ 50

Operations on Radicals ................................................................. 51

Addition .................................................................................... 51


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Multiplication ........................................................................... 51

LINEAR, LITERAL AND WORD PROBLEMS ......................................... 57

Linear Equations and Inequalities ................................................ 57

Literal Equations ........................................................................... 57

Word Problems ............................................................................. 58

Kind of Word Problems ............................................................ 58

QUADRATIC EQUATIONS AND INEQUALITIES .................................. 65

Laws of Quadric Equations ........................................................... 65

Quadratic Formula ........................................................................ 66


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PREFACE

This module entitled Logic of Mathematics is designed primarily

to enrich the psychology of students of Bachelor of Science in Computer

Science enrolled in College Algebra concerning the importance of

studying Mathematics in general and College Algebra in particular. This

module uses several sources to present unquestionable credibility of its

contents. In addition, this module withholds several manipulations in

order to clearly manifest its mission and suit the needs of the students.

Preliminary knowledge about Algebra and Turbo C

Programming Language is necessary to fully comprehend the contentsof this module. Moreover, this module used direct approach in

correlating Mathematics in the Science of Computers. Vivid examples

are provided so that there is enough illustration to present the ideals of

Mathematics in Computer Studies. Furthermore, activities are available

at the end of each chapter to measure and evaluate whether the

student got the necessary understanding before proceeding to the next

chapter.

This module will be the start of your upcoming success in the

field of Computer Science aided by the logic of Mathematics. As what

the Our Lady of Fatima University claims, rise to the top!

James Michael A. Adoremos


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SETS

A set is a well-defined collection of data. The data mentioned

could be anything like objects, numbers, letters and the like repeated or

not. Data belonging to a set are referred to as its elements.

In computer, a set refers to a variable which contains several

values. These are then called array. Arrays are declared like regular

variables just that they have the symbols [] right after the last letter of

the variable name wherein there is a specified limit of maximum values

an array can hold placed in between the symbols []. Elements, on the

other hand, are referred to as its values. It starts its index at zero.

Mathematics:

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

Here a set named A has the elements 1, 2, 3, 4, 5.

Computer:

int A[5] = { 1, 2, 3, 4, 5}

Here an integer array has the values 1, 2, 3, 4, 5.

Sets and arrays are treated the same. Their use and function in

both Mathematics and Computer are similar.

The next lessons on these chapter focuses on Mathematics as they

are not generally used in Computer Programming.


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Kinds of Sets

Finite Sets

Finite Sets are sets in which all of their elements are

enumerated.

Here is an example:

A = { 1, 2, 3, 4 }

B = { x | x is a letter in the word MISSISSIPPI }

The cardinal number of a finite set is the unique countingnumber n such that the elements of a set are in one-to-one

correspondence with the elements of a set of counting numbers from 1

to n. One-to-one correspondence refers to the circumstance where

every element of a set is paired with one and only one element of

another set. It follows the format:

n(NAMEOFSET)Where:

NAMEOFSET is the name of the reference set.

Furthermore, n(A), for example, is read as n of A or the cardinal

number of A. And take note that the cardinal number of a null set is

always zero.

Infinite Sets

Infinite Sets are sets in which not all of the elements are

enumerated.

Here is an example:

A = { 1, 2, 3, 4 }

B = { x | x is a letter in the word MISSISSIPPI }


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Universal Sets

Universal Sets are sets whose elements are the elements of all

of the sets under consideration. Common elements are regarded as

one. This usually uses the name Set U to set distinction from the other

sets.

Here is an example:

A = { 1, 2, 3, 4 }

B = { 4, 3, 1, 2, 1 }

C = { 4, 5, 2, 3 }Set U = { 1, 2, 3, 4, 5 }

Null Sets

Null Sets are sets which does not contain any element. This is

easily identifiable because the set either has no element or has thissymbol .

Here is an example:

A = { }

B = { }

Both sets A and B are null sets.

Relationship of Sets

Equal Sets

Sets are said to be Equal if and only if the sets contain exactly

the same elements without being particular to the arrangement of the

elements or repetition of certain elements. This relationship is shown

through the use of symbol.


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Here is an example:

A = { 1, 2, 3, 4 }

B = { 4, 3, 1, 2, 1 }

C = { 4, 5, 2, 3 }Set A = Set B while neither A and B is equal to C.

Equivalent Set

Sets are said to be Equivalent if and only if the sets contain

equal number of elements regardless whether the elements themselvesare equal. Equivalent sets follow one-to-one correspondence. This

relationship is shown through the use of symbol.Here is an example:

A = { 1, 2, 3, 4 }

B = { 4, 3, 1, 2, 1 }

C = { 4, 5, 2, 3 }

Set A Set C while neither A and C is equivalent to BTake note that all equal sets are equivalent but not all

equivalent sets are necessarily equal.

Joint Sets

Sets are said to be Joint if and only if the sets has at least one

common element.

Here is an example:

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

C = { 4, 5, 2, 3 }

Sets A, B, and C are joint sets because of theircommon elements which are 2, 3 and 4.


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Disjoint Sets

Sets are said to be Disjoint if and only if the sets does not have

any element common to them.

Here is an example:

A = { 1, 2, 3, 4 }

B = { 5, 6, 7, 8 }

C = { 9, 10, 11 }

Sets A, B, and C are disjoint sets because they do nothave any element common to the three of them.

Subsets

Sets are said to be a subset of a set if all the elements of a set

are also elements of another. This relationship is denoted by the symbol

. B A is read as B is contained in A or A contains B where B is asubset of set A.Here is an example:

A = { 1, 2, 3, 4 }

B = { 1, 2, 3 }

Set B is a subset of A , B

A.

C = { a, b, c, d }

D = { d, c, a, b }

Set C is a subset of D , C D, and vice versa.In addition, sets are said to be a proper subset of a set if all the

elements of a set are but some of the elements of another. Therefore, a

set has greater number of elements in comparison to its subset. Thisrelationship is still denoted by the symbol .


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Here is an example:

A = { 1, 2, 3, 4 }

B = { 1, 2 }

Set B is a subset of a , B A.Lastly, sets are said to be a trivial subsets if a set takes into

consideration having a null set as a subset. This relationship is still

denoted by the symbol .Here is an example:

A = { 1, 2, 3, 4 }

B = { }

Set A and Set B, B A, are said to be trivial subsets.

Set Operations

Union

Union is an operation used to combine all of the elements of

the involved sets while considering the common elements as one. This

uses the symbol .Here is an example:

A = { 1, 2, 3, 4 }

B = { 3, 4, 5, 6 }C = A BSet C = { 1, 2, 3, 4, 5, 6 }.

Intersection

Intersection is an operation used to have only the commonelements in the involved sets. This uses the symbol .


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Here is an example:

A = { 1, 2, 3, 4 }

B = { 3, 4, 5, 6 }

C = A BSet C = { 3, 4 }.

Difference

Difference is an operation used to remove the common

elements of the sets involved and disregarding the rest of the elements

of the latter set. This uses the following symbols: /.Here is an example:

A = { 1, 2, 3, 4 }

B = { 3, 4, 5, 6 }

C = A BD = B A

Set C = { 1, 2 } and Set D = { 5, 6 }.

Complement

Complement is an operation used to remove the elements of a

set from the universal set. This is the same as getting the difference of

the Universal Set and the involved set. This uses the symbol .

Here is an example:

U = { 1, 2, 3, 4, 5, 6, 7 }

A = { 1, 2, 4, 5, 6 }

B = A

Set B = { 3, 7 }

Take note that the union of a set and its complement results tothe universal set.


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Set Product

Set Product is an operation used to get all of the possible

ordered pairs where the x coordinate are filled by the elements of the

former set and the y coordinate is filled by the elements of the latter

set. Because of this, this operation is also called Cartesian Productand

Cross Product. This operation uses the symbol x.

Here is an example:

A = { 0, 1, 2, 3, 4 }

B = { 5, 6, 7, 8, 9 }

C = A x BD = B x A

Set C = { (0,5), (0,6) (0,9), (1,5), (1,6) (4,9) } andSet D = { (5,0), (5,1) (6,0), (6,1) (9,4) }

Take not that Set Product is generally not commutative.

Venn-Euler Diagram

Venn-Euler Diagram or simple Venn Diagram is a graphical

representation of data in dealing with the relationship and operations of

sets. It is named after Cambridge Logician John Venn. The Set U is

represented by a rectangle while individual sets are represented by

circles. A shade on a region inside the diagram indicates the relation or

the operation asked.

Here is an example:

U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

A = { 0, 1, 2, 3, 4 }

B = { 0, 4, 5, 6, 7 }

C = { 0, 1, 7, 8, 9 }

What is asked:

A B C


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Explanation:

The example asks for the common element(s) of Sets A,

B and C. This is through the intersection operation. Therefore the

shaded region should be the region where the three involved sets

intersect. And if the problem would ask what elements are contained in

the shaded region, it would be the element 0.


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EXAMPLES

1, Problem:

Create a program which computes for the average of five

numbers which the user will input. Use an array and imposeinteger as the only data type.

Logic:

A user will input one number then the number will be

stored to a variable, which in this case is an array, and then

repeat the process until there are five values. After which,

another variable will compute for the sum of the stored valuesfrom the array then divide it by 5. Then, the result will be

displayed.

Source Code:

#include

#include

main(){

int array[5];

int ctr, ave;

clrscr();

ave = 0;

for(ctr=0;ctr


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Preview:

Review:This program uses an array array[] which will contain

five maximum values. It uses for loop to repeat the value-

storing process. Then the variable ave retrieves the values

stored in array[] and compute for their sum before dividing the

sum with 5 to have the user inputs average.

2, Problem:Create a program that will store names of clients and

their debts. The user will be asked for every entry stored

whether to continue adding data or not. If not, display all stored

clients and their according debts.

Logic:

This program will be almost the same as the one above

just that this will use two arrays and will not compute for

anything. The user will first enter the name of the client which

will be stored in the first array. Then the user will be asked to

enter the debt that client owes. This process will be repeated

indefinitely as long as the user chooses to continue adding a


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data. After which, the stored values will be retrieved from their

respective arrays and displayed back to the user.

Source Code:

#include #include

main(){

char client[106];

int debt[106];

int cont,count,ctr;

clrscr();

cont = 1;

count = 0;

while(cont != 0){

printf("Enter the client's name: ");

scanf("%s",client[count]);

printf("Enter the client's debt: ");scanf("%d",&debt[count]);

count++;

printf("Enter another data

(1==yes,0==no)? ");

scanf("%d",&cont);

}

printf("\n");

for(ctr=0;ctr


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Preview:

Review:

This program stores the clients name under client[]

array and the debts under debt[] array. I set 106 as the

maximum limit because of no particular reason. The variable

cont will dictate whether to continue the program or end it

upon having a value of 0. The variable count served as the

holder of the number of records recorded and was set as 0

because at the start of the program, no record was in place. Iused while loop because the condition is set for a specific

equality value and the loop must repeat unless the cont have a

value of 0. After the while loop, the for loop will retrieve one-at-

a-time the values from the arrays and display it. For loop was

used because the condition applied uses an inequality

statement. The variable ctr served as the counter variable for

the for loop as it reaches the specified condition.


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ACTIVITIES

1, Create a program using arrays which asks the user for the number of

entries to be entered and the values for each entry. Compute for the

average of the values stored and display the average and all repeatedentries.

Sample Preview:


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2, Create a program which will create an arithmetic sequence after

asking the user for the start, end, and gap between the numbers of the

sequence. Display the sequence and ask the user what number to find

its ordinal location in the sequence. Display the ordinal location. Repeat

the searching until the user chooses to stop. End the program with agreeting.

Sample Preview:


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REAL NUMBERS

This chapter will again focus on Mathematics more for only their

logic would contribute to Computer Programming Fundamentals andmathematical computations.

Kinds of Real Numbers

Rational Numbers

Rational Numbers are real numbers that can be written in

fraction form or as a ratio.

Here are some examples:


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Integers

Integers are rational numbers which belong below, above or

exactly at zero on the number line. They are classified into three:

positive, integers above zero on the number line, negative, integers

below zero, and zero which is neither positive nor negative integer.


Natural Numbers

Natural Numbers are rational numbers starting from positive

one, positive two and onwards. Because of this, this set is also called

Counting Numbers.


Whole Numbers

Whole Numbers are rational numbers almost the same as

natural numbers just that this set includes zero as one of its elements.


Fractions

Fractions are rational numbers whose value is either in betweentwo natural numbers or the natural numbers themselves divided by

one. It is written as one of the following forms: ,


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Decimal Numbers

Decimal Numbers are rational numbers almost the same as

fractions just that this set uses the elements of whole numbers and is

written in this form: wholenumber.wholenumber.


Take note that rational numbers in decimal form must never be

non-terminating, non-repeating else the number falls under irrational

numbers.

Irrational Numbers

Irrational Numbers are real numbers if written in decimal form,

the value on the right of the decimal point is non-terminating and non-

repeating. It may be in symbol form, numerical form or decimal form.


Take note that irrational numbers could never be transformed

to fraction form else the number falls under rational numbers.

Properties of Real Numbers

Like everything else, Real Numbers are governed by some

properties. In every property, a translation in symbols where the

variables are all real numbers is presented before the examples.


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Take in consideration some terms used in this chapter for those

will be defined and explained in the next chapter.

Closure

Closure Property states that when real numbers undergo

arithmetic operations, the result would always be a real number.

In symbols:

a + b = c

d * e = f

Where all of them are real numbers

Here are two examples:

1 + 2 = 3

5 * 5 = 25

Where the results are real numbers

Commutative

Commutative Property states that changing the order while

considering their signs will not affect equality.

In symbols:

a + b = b + a

cd = dc


20 + 5 = 5 + 20

4 * 2 = 2 * 4

Associative

Associative Property states that changing the groupings while

taking into consideration the kinds of operation used and the signs of

real numbers will not affect equality.


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In symbols:

a + ( b + c ) = ( a + b ) + c

( d * e ) * f = d * ( e * f )

Here are two examples:7 + ( 10 + 4 ) = ( 7 + 10 ) + 4

( 1 * 0 ) * 8 = 1 * ( 0 * 8 )

Distributive

Distributive Property states that when a quantity is multipliedto a grouped quantity, the quantity is then multiplied to the individual

members of the grouped quantity which are separated by addition or

additions inverse.

In symbols:

a * ( b + c ) = a * b + a * c

d * ( e * f ) = d * e * f


2 * ( 5 + 3 ) = 2 * 5 + 2 * 3

7 * ( 0 * 23 ) = 7 * 0 * 23

Identity

Identity Property states that when a quantity is multiplied to its

identity element, the result would always be the quantity itself. The

identity element of Addition is zero while of Multiplication is one.

In symbols:

a + 0 = a

b * 1 = b


99 + 0 = 99

41 * 1 = 41


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Inverse

Inverse Property states that when a quantity undergoes an

arithmetic operation wherein the other quantity involved is its inverse,

the result would be its identity element which depends on the kind of

arithmetic operation used. It is zero in addition and one I multiplication.

The inverse of a quantity undergoing addition is the negative of the

quantity. The inverse of a quantity undergoing multiplication is its

reciprocal.

In symbols:

a + ( -a ) = 0b * (

) = 1


3 + ( -3 ) = 0

6 * ( ) = 1

Reflexive

Reflexive Property states that a quantity is always equal to

itself. It is usual practice that after this property is used in Mathematics

transitive property follows.

In symbols:a = a

Here is an example:

20 = 20


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Symmetric

Symmetric Property states that interchanging the left side and

the right side of an equation does not affect its equality. This is usually

used if, in an equality where both side is of only one quantity, the

variable is located on the right side and its constant is on the left side of

the equality.

In symbols:

a = b

b = aHere is an example:

3 = x

x = 3

Transitive

Transitive Property states that if quantity A is equal to quantity

B and quantity B is equal to quantity C, then quantity A is equal to

quantity C.

In symbols:

If a = b

And b = c

a = cHere is an example:

If d = 9

And 9 = e

a = e


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Substitution

Substitution states that if there exists an equation and a

quantity involved in the said equation is equal to another quantity, then

replacing the former quantity in the said equation with the latter

quantity will not affect equality.

In symbols:

If a + b = c

And a = d

d + b = cHere is an example:

If a + 9 = 20

And a = 11

11 + 9 = 20

Addition

Addition Property states that in an equation, adding a quantity

to both sides will not affect equality.

In symbols:

If a + b = c + d

Then e + ( a + b ) = e + ( c + d )

Here is an example:

If f + 11 = g + 9

Then h + ( f + 11 ) = h + ( g + 9 )

Take note that a method called transposition is just an addition

property shortcut wherein a quantity is removed from one side of the

equation then the inverse of the quantity is added to the other side ofthe said equation. Also take note that subtraction is only an inverse of

the addition process thus subtraction itself is addition of a negative

value.


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Multiplication

Multiplication Property states that in an equation, multiplying a

quantity to both sides will not affect equality.

In symbols:

If a + b = c + d

Then e * ( a + b ) = e * ( c + d )

Here is an example:

If f + 7 = g + 3

Then h * ( f + 7 ) = h * ( g + 3 )

Take note that division is only the inverse of multiplication.

Thus, dividing a quantity by a number is simply multiplying the inverse

of the number to the said quantity.


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EXAMPLES

1, Problem:

Create a program that will reverse the values of two

variables. Display the former and new value of the twovariables.

Logic:

Two variables will store the entered data by the user

respectively. A third variable, acting as a temporary data holder

or variable, will equate to one of the two variables. The one

chosen between the two variables will equate to the not chosenvariable. The not chosen variable will then equate to the third

variable.

Source Code:

#include

#include

main(){

char var1, var2, var3;

clrscr();

printf("Enter data1: ");

scanf("%s",var1);

printf("Enter data2: ");scanf("%s",var2);

printf("\nBefore:\n");

printf("\tdata1: %s\n",var1);


var3=var1;

var1=var2;

var2=var3;

printf("After:\n");


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getch();

}

Preview:

Review:

The program uses a simple logic in comparison to the

Real Numbers especially its properties. The logic itself explained the

code used.

2, Problem:

Create a program which will compute for the depart

value, arrive value, and days of recovery basing from the users

entered values which are the hours traveled, time zones

crossed, departure time and arrival time. Consider the following

information:


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>>Depart value = {

0, if departure time is between 8am and 12pm.

1, if departure time is between 12pm and 6pm.

3, if departure time is between 6pm and 10pm.4, if departure time is between 10pm and 1am.

5, if departure time is between 1am and 8am.

}

>>Arrive value = {

4, if departure time is between 8am and 12pm.



1, if departure time is between 10pm and 1am.

3, if departure time is between 1am and 8am.

}

>>Days of recovery =

( hours_traveled / 2 + ( zones_crossed 3 ) +depart_value + arrive_value ) / 10

>>All of the time should follow the 24-hour format.

This problem was taken from OLFU 2011 CSIT Week Computer

Programming Tournament Difficult Round hence do I not claim this as

mine.

Logic:

This problem will just need conditionals to get the

depart and arrive values and a simple computation for the days

of recovery.

Source Code:

#include

#include

main(){


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float hours, zones, dt, at, depart, arrive, dor;

clrscr();

printf("Enter hours traveled: ");

scanf("%f",&hours);printf("Enter time zones crossed: ");

scanf("%f",&zones);

printf("Enter departure time: ");

scanf("%f",&dt);

printf("Enter arrival time: ");

scanf("%f",&at);

if(dt>8&&dt12&&dt18&&dt22&&dt8&&at12&&at18&&at22&&at


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Preview:

Review:

Here, we asked the user to enter the necessary

information to compute for the days of recovery. If conditional

was used to determine the value of depart and arrive

individually. The computation for the days of recovery follows

the formula given just that the variables contain the appropriate

values for an accurate computation.


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ACTIVITIES

These activities were also taken from OLFU 2011 CSIT Week

Computer Programming Tournament Easy and Average Roundrespectively.

1, Problem:

Create a program which would compute for the total

payable by the user after ordering a product. The user will enter

the following information: product name, product price, and

overnight delivery confirmation. Let the product price be incentavos. Shipping charge is 2 pesos for items worth 10 pesos

below and 3 pesos for the rest. An additional 5 pesos will be

charged of the user agreed for an overnight delivery. Have the

user apply necessary spaces.

Sample Preview:


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2, Problem:

Create a program which will categorize the cash amount

the user entered as 1000, 500, 100 and 50 bills. Display the

remaining bills. Consider and display only the least possible

combination.

Sample Preview:


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ALGEBRAIC EXPRESSIONS

This chapter will again focus on Mathematics more for only their

logic would contribute to Computer Programming Fundamentals,

mathematical computations and data manipulations.

Polynomials

Some Terms:

Variables are one-character letters which is used as a

temporary representation of an unknown in order to find a value for theunknown. This usually uses lowercase letters. And the most used

variable of all are x and y.


Let a = age

b = time

x = interesty = principal

Constants are the numbers itself without any variable being

multiplied to it.


5, 20, 47, 90

Terms are a number, a variable or a combination through

multiplication of number and variables in an expression. They are

separated in by the addition. In a term, the numbers are called


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numerical coefficients and the variables are called literal coefficients.

Similar terms, commonly known as like terms, are terms having the

same literal coefficients.

Here are some examples:One term: 7x

Two terms: 9y + 3x

Three terms: 3xy + 17a 30b

Base is the quantity, either a constant or variable, which is

raised to a certain degree.


In 59, x0, t2, 3m,

The bases are 5, x, t and 3.

Exponent is the quantity same as base just that this speaks of

the degree itself.

Here are some examples:In the examples on base,

The exponents are 9, 0, 2 and m.

Algebraic Expressions are a term or combination of terms such

that there is defined or finite number of involved arithmetic operations

like addition and multiplication. There are 4 kinds of algebraic

expressions depending on the number of terms used. They are asfollows: monomial with one term, binomial with two terms, trinomial

with three terms and multinomial with more than three terms.


x, 2, 51p, ( 2y + 5z)2, ( x2 y2 ), m-3 n

-3

Polynomials are an expression similar to algebraic expression

just that this does not include a negative exponent raised to any of its

terms, numerical coefficients, literal coefficients or constants. The

degree of a polynomialis determined by the highest exponent one of its

terms has.


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X2 + 3x 4, a + b3, u +2t, p 6

Operations on Polynomials

Addition and Subtraction

In adding polynomials, similar terms are the only ones

to be evaluated. Grouping the similar terms might prove to be

useful in evaluating expressions but take note that such apractice will take more time. The rest are retained as is. Order is

not important. The usual practice is to arrange the evaluated

expression by descending powers of a certain variable. In

subtracting polynomials, it might be useful to change the

subtraction symbol to addition and negate the term changed. In

such a way, the only arithmetic operation to be performed is

addition.


1, ( 4a + ( 9x + 2 ) ) + ( 3a + ( 1 + 2x ) )

= 4a + 9x + 2 + 3a + 1 + 2x

= ( 4a + 3a ) + ( 9x + 2x ) + ( 2 + 1 )

= 7a + 11x + 3

2, ( 6s + 5y ) + ( -5x - 2y + 4 )

= 6s + 5y + (-5x) 2y + 4= 6s + ( 5y + ( -2y) ) 5x + 4

= 6s +3y -5x + 4

Multiplication

Laws of Exponents:

1, an * am = an + m

2, ( an )m = anm


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3, = a

( n + (-m) )

4, ( a * b )n = an * bn

5, ( )

n =

Monomial Monomial:

In monomial to monomial occurrence, simply multiply

the two while considering the laws of exponent and the

properties of real numbers.

Monomial Polynomial:

In monomial to polynomial occurrence, use the

distributive property of real numbers to each of the terms of

the polynomial. Multiply them like monomial to monomialoccurrence. If there exist similar terms, apply the indicated

operation until there exist no similar terms.

Polynomial Polynomial:

In polynomial to polynomial occurrence, distribute first

the first term of the first polynomial to the terms of the otherthen the second term to the terms of the other until all of the

terms of the first polynomial has been distributed to the terms

of the other polynomial. Multiply them like monomial to

monomial occurrence. And if there exist similar terms, apply the

indicated operation until there exist no similar terms.

Special Products

Here are the most of the special products used in Algebra:

01, a( b + c + d + e ) = ab + ac + ad + ae


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02, ( a + b )( a b ) = a2 b

2

03, ( a + b)2 = a2 + 2ab + b2

04, ( a b)2 = a2 2ab + b

2

05, ( a + b )( a2 ab + b2 ) = a3 + b3

06, ( a - b )( a2 + ab + b2 ) = a3 - b3

07, ( x + a )( x + b ) = x2

+ x ( a + b ) + ab

08, ( ax + b )( cx + d ) = acx2 + x ( ad + bc ) + bd

09, ( a + b + c )2 = a2 + b2 + c2 + 2ab + 2bc + 2ac

10, ( a + b )3 = a3 + 3a2b + 3ab2 + b3

11, ( a b )3 = a3 - 3a2b + 3ab2 - b3

Binomial Theorem

This theorem states that for any term (a+b) raised to a

power n,

( a + b )n = ( )a

n + ( )a

n-1b + ( )a

n-2b2 + + ( )b

n

Where () =

Pascal Triangle

The Pascal triangle is a triangular illustration to show

the numerical coefficients of a binomial in expanded form if

raised to a certain number. This does not show the powers by


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which the variables are raised to. But it is thought that the

power of a variable a is decreasing until it reached a power of 0

while for variable b, it is increasing. Here is a part of the well-

known Pascal Triangle.

BINOMIAL PASCAL TRIANGLE EXPANDED FORM

( a + b )0 1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1

( a + b )1 a + b

( a + b )2 a2 + 2ab + b2

( a + b )3 a3 + 3a2b + 3ab2 + b3

( a + b )4 a4 + 4a3b + 6a2b2 + 4ab3 +b4

This sequence continues onwards. The numerical coefficient of

a term, as seen in the diagram, is the sum of the two numerical

coefficients above it. Let us take 4th row as an example. The second

numerical coefficient which is 3 is the sum of 1 and 2 which are above it.

Factoring

Factoring is changing an expression into a product of at

least two factors.

Types of Factoring

1, Common Factor it is factoring an expression by a

quantity present to the terms of the expression.

Here is an example:

ab + ac = a( b + c )

2, Inverse of Special Products it is factoring using the

concepts of special products. It is reversing the process

undergone when factors are expanded using special products.

Here is an example:

x2 y2 = ( x + y )( x y )


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EXAMPLES

1, Problem:

Create a program which will display an equiangulartriangle of asterisks. The asterisks must start with one asterisk

at the first row and incrementing in number by one asterisk as it

goes to the next row. The number of rows of asterisks will be

determined by the user.

Logic:

The last row of the asterisk is expected to touch the side

of the DOS window therefore spaces are present before the first

asterisk of each row. The first row will have n-1, where n is

equal to the user-entered value, spaces before the asterisk. The

next will be n-2 and so on. The 1st row will have 1 asterisk, the

2nd with two, the 3rd with 3, and so on.

Source Code:

#include

#include

main(){

int rows;

int ctr, spaces, asterisk;clrscr();

printf("Enter the number of rows to be displayed: ");

scanf("%d",&rows);

printf("\n");

for(ctr=1;ctr=ctr;spaces--){

printf(" ");

}

for(asterisk=1;asterisk


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printf("* ");

}

printf("\n");

}

getch();

}

Preview:

Review:

The program uses four variables. The rows variable

stores the number of rows of asterisks asked by the user. The

ctrvariable stores the current row that the program is working

on. The spaces stores the number of spaces left to be displayed

for the current row the program is working on. The asterisk

stores the current number of asterisks displayed. The ctr for

loop works for each row. The spaces for loop works for the

spaces and the asterisk for loop works for the asterisks to be

displayed. And before the ctr for loop ends, a next line

statement is displayed for the program to work on the next row.


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2, Problem:

Create a program similar to the one right before this

just that the figure to be created would be a diamond-likerhombus. Other conditions remain the same. The total number

of rows would be the twice the value entered by the user minus

one.

Logic:

The logic of this program is almost to the one before

this with additional looping for the inverted triangle having a

deduction of one row to the number of rows of the original

triangle.


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Source Code:

#include

#include

main(){

int rows;

int ctr, spaces, asterisk;

clrscr();

printf("Enter a whole number: ");

scanf("%d",&rows);

printf("\n");

for(ctr=1;ctr=ctr;spaces--){

printf(" ");

}

for(asterisk=1;asterisk0;ctr--){

for(spaces=ctr;spaces


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Preview:

Review:

The source code of this program is almost similar to the

one before it. An additional ofctr for loop is present after the

first one. This works on the inverted triangle. The conditions

and initializations of the inverted triangle except for the asterisk

for loop are somehow inverted to suite the problem. The

asterisk for loop retained the same for the reason that theprocess of displaying the asterisks is just repeated. The reason

why the asterisks in the 2ndasteriskfor loop decreases because

it is dependent on the 2nd ctr for loop which was modified to

decrease its value instead of increasing it.

The examples seem not connected to Algebraic Expressions. They

show how data would be manipulated during the duration of program

execution. This is very evident. Logic contained in those examples is of

greater importance in compared to number manipulations alone!


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ACTIVITIES

1, Create a program which would display a triangle having n rows,

where n is the whole number entered by the user. The triangle must be

composed of whatever the user entered as characters to be displayedwhich may contain spaces.

Sample Preview:


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2, Create a program which would display a diamond-like rhombus

having n rows, where n is a whole number entered by the user. The

figure must be composed of but asterisks and spaces only. The number

of allowable rows is an odd number starting from 3. If the user entered

an even number, deduct one to his entered value. Disallow enteredvalues lower than 3. Display on the sides the current row number of the

row it belongs to.

Sample Preview:

In order to solve the activities above within this chapter, the logic

within each problem must be well understood and algebraically

formulated. After which, data manipulation will come. Only after then

would these activities be solved accurately and properly.


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RADICALS

This chapter will again focus on Mathematics more for the only

application of this lesson in Computer Programming is on data

manipulations and mathematical evaluations.

If a variable a and another b are real numbers and n as a

positive integer greater than 1 such that bn = a, then b is called the nth

root of a. The nth root of a is written as in symbols. If the n is 2, it is

referred to as square and cube for n having he value of 3.

Here are examples:

( 9 )3

= 27, 9 is the cube root 27.( 2 )5 = 64, 2 is the 5th root of 64.

( 11 )2 = 121, 11 is the square root of 121

The entire expression is called a radical. The symbol isthe radical symbol. The n is called the index or the order of the radical.

The quantity inside the radical symbol is called the radicand. The default

index of a radical is 2 or square thus it is customary to be omitted.Radicals are said to be similar if and only if their indexes and radicands

are equal.

Converting a radical into an equivalent expression wherein

there is no radical in the denominator, if written in fraction form, is

called rationalizing the denominator.

Laws of Radicals

1, , if is a real number.


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2,

3,

4,

Operations on Radicals

Addition

Radicals could undergo addition, or subtraction, if and

only if they are similar radicals. They are added, or subtracted,

by undergoing the inverse of distributive property of real

numbers on the appropriate coefficients of the terms before

evaluating the coefficients.

Here is an example:

Multiplication

There are two ways to multiply, or divide, radicals

depending on the indexes of the radicals.

Similar Indexes

To multiply, or divide, radicals of similar

indexes, simply multiply the coefficients of the radicals

then multiply the radicands of the radicals. Of course,

simplify the result.


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Here is an example:

Different Indexes

To multiply, or divide, radicals of different

indexes, convert the complying radicals such that they

will have the same indexes. Those which could not be

converted will be retained as is. After, apply the methodof multiplying radicals having similar indexes.

Here is an example:


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EXAMPLES

1, Problem:

Create a program which will solve a one term radicalexpression the user will define. Thus ask the user for the base

and the exponent of the radical.

Logic:

This program is very basic such that the program will

only substitute the values entered by the user to a defined

expression and evaluate the expression to show the exactanswer.

Source Code:

#include

#include

#include

main(){

int base, exponent, answer;

clrscr();

answer = 0;

printf("Enter the base: ");scanf("%d",&base);

printf("Enter the exponent: ");

scanf("%d",&exponent);

answer += pow(base,exponent);

printf("\nThe answer is %d.",answer);

getch();

}


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Another Source Code:

#include

#include

main(){

int base, exponent, answer;

int ctr;

clrscr();

printf("Enter the base: ");

scanf("%d",&base);

printf("Enter the exponent: ");

scanf("%d",&exponent);

answer = base;

for(ctr=2;ctr


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Preview:

Review:

Both of the source codes above will display the same

output and undergo the same process. The first code just uses

the math library to solve the radical. The second uses for loop

with its variable ctr. Both will have the base multiplies

repeatedly to itself until it satisfy the intended condition.


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ACTIVITY

1, Create a program which will solve a one term radical expression

the user will define. Thus ask the user for the coefficient, the index, and

the radicand of the radical. Just round the final answer to the nearesthundredths.

Sample Preview:


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LINEAR, LITERAL AND WORD PROBLEMS




Linear Equations and Inequalities

An equation is a statement showing equality between two

expressions. An inequality refers to a statement showing that one of the

two expressions involved is higher or not equal to the other. Both followsome of the properties of real numbers. They are as follows: Reflexive

property, Symmetric property, Transitive property, Addition property

and Multiplication property.

The degree of a term refers to the sum of the exponents of the

literal coefficients or variables.

An equation is said to be a linear equation or linear inequality if

its terms are raised up to but the first degree.

Here is an example:

x 2y + z +a = b + d n

Literal Equations

A literalequation is a statement of equality where some if not

all of the unknowns are represented by a letters or the variables. The

widely used examples of literal equations are the formulas.


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Here is an example:

Word Problems

Word Problems are situational-based problems which involves

computations to find a certain unknown. Translation refers to the

transformation of a phrase in the word problems in order to create an

expressions, equalities or inequalities. The equations used in word

problems are called mathematical models.

Kind of Word Problems

1, Number Relation Problems are problems which

concerns the relationships among integers, fractions,

percentage and the like.

2, Age RelatedProblems are problems concerning the

number of years or age as the unknown.

3, Work Problems are problems concerning the time

needed to finish a job and the rate a person finishes a job. This

may involve more than one persons involvement.

4, Mixture Problems are problems concerning with

solutions or mixtures, their percentage of concentration and the

amount of materials involved.

5, Motion Problems are problems concerning basic

physics such as time, velocity, distance and the like.

6, Mensuration Problems are problems concerninggeometric figures like triangles, rectangles and circles.

7, Investment Problems are problems concerning

income, rates, investments, principals, and the like.


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EXAMPLES

1, Problem:

Jericho owns a big house. He is going to place marbletiles as his flooring. He only knows the area of the tiles he will

need. Before he buys the tiles, he wants to verify if the tiles he

will buy would be enough, would be lacking or would be

exceeding what he needs. Create a program to solve his

problem. Note that the tiles are of the same sizes and he will

enter the land area of his house, the number of tiles he is

planning to buy and the area of the tiles.

Logic:

The program will just compute for the product of the

area of the tiles and the total number of tiles he will buy and

compare the product to the land area.

Source Code:

#include

#include

main(){

float land, tile, num;

clrscr();

printf("Enter land area: ");

scanf("%f",&land);

printf("Enter tile area: ");

scanf("%f",&tile);

printf("Enter number of tiles: ");

scanf("%f",&num);

printf("\n");

if(land==tile*num)

printf("The tiles will be enough for the house.");


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else if(land>tile*num)

printf("You will lack tiles.");

else

printf("You exceeded the needed number of

tiles you\'ll need.");

getch();

}

Preview:

Review:

The program looks and works in basic structure. No

much reviewing is needed. Just that the program records the values

entered by the user and compares the product concerning the tiles with

the land area and display appropriate terms.


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2, Problem:

Jericho is now problematic because he could not

anymore think how large and how many tiles he will need for

his floor house. Create a program to solve his problem. Notethat the program would be suggesting how much is still needed

and also will be suggesting the area of tiles are appropriate if his

entered area is either would exceed or lack the number he will

be needing.

Logic:

This program will just be the same like the one before

this just that the program will compute for the lacking or excess

tiles and if needed would compute for the appropriate tile area

and how many he will need.


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Source Code:

#include

#include

float approp(float x, float y){

float exact;

exact=x/y;

return exact;

}

void main(){

float land, tile, num;

clrscr();

printf("Enter land area: ");

scanf("%f",&land);

printf("Enter tile area: ");

scanf("%f",&tile);

printf("Enter number of tiles: ");scanf("%f",&num);

printf("\n");

if(land==tile*num){

printf("The tiles will be enough for the

house.");

}else if(land>tile*num){printf("You will lack %.0f tiles.",land-tile*num);

printf("\nWhy not buy %.0f tiles with an area

of %.0f.",num,approp(land,num));

}

else{

printf("You exceed %.0f tiles.",tile*num-land);

printf("\nWhy not buy %.0f tiles with an area

of %.0f.",num,approp(land,num));

}

getch();

}


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Preview:

Review:

The program uses another function named approp

which would compute for the suggested tile area depending on

the land area and the number of tiles the user plans to buy. Theother things are just the same with the program before this. An

additional line which would display the computed suggested tile

area if the land area is lesser or higher to the product of the tile

area the user entered and the number of tiles.


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ACTIVITIES

1, A student from Pinalpal Elementary is academically

challenges when speaking of Mathematics. He does not know

how to properly solve a simple equation.

Create a program that will help him compute for the

value ofx. Just provide him a formula to base with and ask for

the needed values or numerical coefficients and constants.

Sample Preview:


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QUADRATIC EQUATIONS AND INEQUALITIES




A Quadratic Equation is an equation having a variable to the 2nd

degree at most. ax2 + bx + c = 0 is the so-called standard form of a

quadratic equation. Quadratic Inequality, on the other hand, shows

inequality between the left side and the right side of the inequality.

The roots of a quadratic equation are the solutions or values ofa variable making the equation valid and true. Solving a quadratic

equation means solving for the value of the unknown.

Laws of Quadric Equations

1, Zero Product Property If the product of two quantities is

zero, then at least one of them is zero.


1, xy = 0;

2, ( x -2 ) ( y + 9 ) = 0


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2, Square Root Property A number of the 2nd degree always

has two square roots. Either both are positive, both are negative or

one is negative while the other is positive.

Here are some examples:1, ( )( )

( )( ) ( )( )

2, ( )( ) ( )( )

( )( )

Quadratic Formula

This is a formula derived from the standard form of a quadratic

equation which solves the roots of the said equation directly. Here is the

formula:

The only step left to do is to substitute the coefficients of the

variables written in the standard form as replacement to the

corresponding variable in the formula.

Here is an example:Given the equation, 9x2 + 25x + 49 = 0, solve for x.


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EXERCISE

1, Problem:

You got tired of manually computing for the roots of x ina provided quadratic equation. Since you are a programmer,

you thought of an idea.

Create a program which will let you enter the values of

a, b and c. Afterwards, the program will compute for the roots

of x.

Logic:

The program will just be simple as it seems. Though the

values ofxmay not be very accurate

Source Code:

#include

#include

#include

main(){

float a, b, c, x1, x2;

float radical;

clrscr();

printf("Enter a: ");

scanf("%f",&a);

printf("Enter b: ");

scanf("%f",&b);

printf("Enter c: ");

scanf("%f",&c);

radical = pow((b*b)-(4*a*c),.5);

x1 = (radical-b) / (2 * a);


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x2 = (radical+b) / (2 * a);

printf("\nx1 = %f",x1);

printf("\nx2 = %f",x2);

getch();

}

Preview:

Review:

The program simply computes for the values of x

through the quadratic formula in linear form. And because

there are two expected answers, both of them were shown

whether or not they are equal.


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ACTIVITIES

1, Another student hates Physics. For all she cares, shell

choose Music over Science! But hey, you have a crush on her.

And you surely would like to help her.

Create a program which would solve the given formula

in Physics. Let the user enter whatever information regarding

the formula and supply you with the values. You have to

identify what quantity is unknown and solve for it.

The Formula:

Sample Preview:

Logic of Mathematics (College Algebra for Computer Science)

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Transcript of Logic of Mathematics (College Algebra for Computer Science)