Matematika Diskrit : Relasi, Sifat Relasi, dan Contoh Soal

Pemodelan untuk Komputasi : Algoritmic Thinking atau Berpikir Secara Algoritmik, beserta Contoh Soal ( Bahasa Inggris )

Introduction of Problem Solving

In the absence of computer 

In the presence of computers

• How can computers solve human problems? 
• More essential question : How can computers understand human problems? 
• Two approaches : programming + intelligent systems 

Software Development Process

Algorithms

The origin of the word "Algorithm"is came from 9th Century Persian Mathematician, Abu Abdullah Muhammad ibn Musa Al-Khwarizmi. Algorithms is a very basic and (probably) the most important thing on the world of computer. It is a well-defined computational procedure to solve computational problems. It takes some value as input. It produces some value as output. 

Input and Output

• Input : Data or values to be processed by algorithm in order to solve the problem
• Output : data or values that are produced by the algorithm in order to solve the problem

Data

There are so many kind of data. Based on the values that are contained by the data, we can classify data as:

• Numeric data : measurable (real number, integer, etc) 
• Example : time, height, weight, amount, etc 

• Non-numeric data : qualitative data, letters, symbol, true or false 
• Example : course name, course code, marital status, etc

Variable

Sometimes, we need to store our data. When we do this, we need some variable. For example:

  Adding two numbers 

• Input: 
• The first number, an integer, stored in a variable called firstNumber 
• The second number, an integer, stored in a variable called secondNumber 
• You can calculate the result of addition operation and store it as new data into a variable called result. 

• Output: result

Example of Algorithm

The Area of Circle

The Glass Problem

Womoes has 2 children. His first child is Womorange and the second one is Womberry. Womorange doesn’t want to drink anything but orange juice from his blue glass. Womberry doesn’t want to drink anything but strawberry juice form his transparent glass. On Monday before school, Womoes put the orange juice in transparent glass and strawberry juice in the blue glass. Help Womoes to swap the juices! They only have one blue glass and one transparent glass, but they have unlimited number of glasses with any kind of colors.

Exercise 1

For each problem below, write the algorithms to solve the problem! 

• In 2018 you save your Rp x in a bank and you get 5% interest each year, in what year will you have more than Rp 2x? You never withdraw your money after the first time you save it. 

• Wobung learns a new language. Every i-th day, he learns i number of new words. How many words will he learn after a year?

• Womfarmer has a garden. The garden is a rectangle-shape garden. He divided the garden into two parts as shown below. Womfarmer plants sunflower in circle part and rose in blue part. If the width, the length of garden, and the radius of circle area are given, write the algorithms to find the area of blue part ! 

Flowchart

Type of diagram that represents an algorithms, workflow or process. It shows the steps as boxes of various kinds, and their order by connecting the boxes with arrows. It illustrates a solution model to a given problem. They are used in analyzing, designing, documenting or managing a process or program in various fields.

The two most common types of boxes in a flowchart are :

• A processing step, usually called activity, and denoted as a rectangular box. 
• A decision, usually denoted as a diamond.

 

Flowchart: Some Building Blocks and Fundamentals

Flowchart Example: Addition

Exercises 2

Write the algorithms and the flowchart to solve each problem below: 

• A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. What is the speed of the car ?

• Three families share a garden. They usually clean the garden together at the end of each week, but last week, family C was on holiday, so family A spent 5 hours, family B spent 4 hours and had everything done. After coming back, family C is willing to pay $90 to the other two families. How much should family A get? You may assume both families were cleaning at the same speed. 

• Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are… 

• Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? 

• What is the angle between the minute hand and the hour hand of a clock when the time is 4.20? 

• The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child? 

• What least number must be added to 1056, so that the sum is completely divisible by 23 ? 

• Today is Monday. What day will it be after 61 days? 

• The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number? 

• We have three different integers, x, y and z, You are asked to find the values for x, y and z for given values of A, B and C which satisfy the following three relations: 
• x + y + z = A 
• xyz = B 
• x^2+y^2+z^2=C 

• A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man's eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60º. What is the distance between the base of the tower and the point P? 

• Two numbers are respectively 20% and 50% more than a third number. What is the ratio of the two numbers? 

• A, B, C subscribe Rs. 50,000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35,000, how much does A receive?