Definition of Sets to History, Ways of Expressing, and Types

Definition of Set – When talking about “set”, what does Sinaumed’s immediately think of? What about calculating sets in mathematics? Or even about the organization of activities in the college major? Even though they look different, both are the same thing, that is, both are in the form of a “group” or “assemblage” that houses objects in a clear definition. Well, this time we will discuss sets in the realm of mathematics

This set material will generally be taught by teachers since we enter junior high school, in grade 7 to be precise. That is why, set material will be related to Algebra which is equally studied in grade 7 junior high school. So, what is this set? What types of sets are often studied? What about the relations between sets? So, so that Sinaumed’s understands it, let’s look at the following review!

What is the definition of a set in mathematics?

In general, the definition of a set can be interpreted as a collection of objects which have clear definitions and can be differentiated. In short, a set becomes a “collection” of well defined objects .

The term “well-defined” means that for any given object X, we will always be able to determine whether the object belongs to a certain set or not. Then, why is a clear definition needed? This is so that people can determine whether the object or thing in question is a member of the intended set or not.

Well, the objects contained in the set are referred to as “elements”, “elements”, or “members”. Objects that can be included in a set must have certain characteristics in common. The same certain characteristics must also be defined precisely , so that Sinaumed’s can also collect them into a suitable set.

According to Hambali and Siskandar (2002:1), the limit of this set is a collection of objects that are real or not real. For example, a herd of horses, a group of letters, a group of chickens. So, from this example, the words “a flock”, “a group”, or “a group” are the same as a set. Other terms for sets are groups, clusters, families, classes, sets, and others. In order for Sinaumed’s to better understand what a set is, consider the following example

Examples of Sets and Non-Sets

Example Set:

1. Collection of Four-legged Animals

“Four-legged Animals Group” is a collection. So, if there is a group of animals that are: spiders, ducks, whales, cats, buffaloes, dogs), then Sinaumed’s must already know which animals have 4 legs. Yep, that’s right, there are cats, buffaloes, and dogs.

Meanwhile the rest, namely: spiders, ducks and whales, are not members of the group of Quadrupeds. Moreover, this set of Quadrupeds is very clearly defined.

2. First Set of Four Even Numbers

“First Set of Four Even Numbers” is also a set. The set contains several clearly defined objects, namely the numbers 2, 4, 6, 8, and so on.

Non-set Example:

  • Collection of numbers
  • Beautiful collection of paintings
  • Collection of delicious food

Why can’t the above examples be called sets when it is clear that these examples are collections that have objects? This is because the object is very abstract. That is, the objects in the set are so abstract that one can only think about them, cannot be seen, felt, touched, or touched.

In the first example, the “Collection of Numbers” has objects that are numbers and are very abstract. Yep, the number object is not certain so we also cannot determine what numbers are included in the set. Given that there are many kinds of numbers, right?

Then in the second and third examples, the objects are paintings and food, respectively, which are concrete objects. However, the two objects are also not certain , you know , because beautiful and delicious properties are relative.

So, it can be concluded that

“A set is a collection of objects or objects that can be clearly defined”. 

A Brief History of the Set Concept

Historically, the theory of mathematical sets has become known to the general public since the late 19th century. But at that time, the emergence of the concept of a set was still being debated among mathematicians. Finally, in 1920 AD, the concept of this set became a subject of discussion in the field of mathematics. The person who introduced the concept of this set was Georg Cantor , a German mathematician. For the concept he initiated, he earned the nickname the Father of Association. This is because he was the first figure to develop the existence of set theory to infinite set theory.

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The father of this Association was born with the name Georg Ferdinand Ludwig Philipp Cantor on March 3, 1845 in St. Petersburg, Russia. During his primary education, he did not go to school like ordinary children, but used private tutors. Then at the age of 11, he moved to Germany with his family. Right in 1873, at the age of 28, he announced a theory through a letter, even for 10 years.

Although set theory and the concept of infinite numbers rocked the world of mathematics, he still did not gain any benefit from these discoveries. After that, around 1867-1871, he published several articles which contained problems with number theory as a continuation of his set theory. It was only at the end of the 19th century that his theory had a major influence on the mathematical literature.

In his ideas, he stated that this set is ‘a collection of objects that have certain and clear conditions’. Objects can also be objects, numbers, or anything which is then referred to as an element or member of a set. Now, the elements of a set must be clearly defined, because to distinguish which are members of the set, and which are not members of the set.

His most influential writing is the concept of infinite sets, published by Crelle’s Journal in 1874. For this innovation, he was finally recognized as the Father of Sets and later died on January 6, 1918.

Set Notation and How to Express It

Set Notation

Basically, the term “set” has a special sign notation, which is in the form of curly brackets like { }. Usually, this set will also be named using capital letters , for example A, B, C, X, and others. While the use of lowercase letters is used to enrich the members of the set .

Membership of a set is expressed by a symbol in the form of , which is read as “member of” . Meanwhile, to declare a member that is not included in the set, it will be denoted by the symbol which reads “not a member of” . Example:

There is a set A which is defined as the set of rainbow colors. The correct answers regarding set A are orange, red, blue, green, yellow, indigo, and purple. So, from this statement, it can be denoted as: Set A = {orange, red, blue, green, yellow, indigo, and purple}

Meanwhile, the membership can be written as:

Orange ∈ A

Red ∈ A

Green ∈ A

Yellow ∈ A

etc

If there is an answer stating that black is included in membership A, then that answer is clearly wrong, right ? So from being declared as black ∉ A, it means that black is not a rainbow color, aka a member of set A.

How to Declare Sets

According to set theory, there are 3 ways that can be used to express sets, namely in the form of tabulations, notation for forming sets, and by mentioning the terms of membership. Well, here is the explanation.

a) Tabulation alias Register ( The Roster Method )

Through this method, later we are required to mention or register the members of the association one by one. In writing, it must be separated by a comma (,) yes… . Please note that in writing the members of this set, it must be clear and nothing should be repeated. Suppose {a, a, b, c, d, d, d}

So, here is an example of writing a set using the tabulation method:

  • Set B is a set of vowels. Then it can be written as: B = { a, i, u, e, o }
  • Set A is the set of natural numbers less than 9. Then it can be written as: A = { 1, 2, 3, 4, 5, 6, 7, 8 }
  • Set K is a set of provincial capitals on the island of Java. Then it can be written as: K = { Jakarta, Serang, Bandung, Semarang, Yogyakarta, Surabaya }
  • The set D is the set of negative integers that are more than 10. Then it can be written as: D = { -9, -8, -7, -6, -5, -4, -3, -2, -1 }

b) With Set Forming Notation ( The Rule Method )

Through this method, later the members of the set will be declared with variables (substitutes or modifiers), which are then followed by dashes, and continued by mentioning the properties or characteristics of the elements of the set. For example,

  • C = { x | x wind instrument }

Read: set C is the set x such that x is a wind instrument.

  • P = {x| x is a prime number less than 12}

Read: P is a set with x such that x is a prime number less than 12.

  • L = {x| x names of districts/cities in Central Java Province}
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Read: L is a set with x such that x is the names of districts/cities in the province of Central Java.

  • D = {x │ x are the first five letters of the Latin alphabet}

Read: the set D is the set x such that x is the first letter of the Latin alphabet.

c) By stating the terms of membership

Through this method, later the members of the set will be expressed by means of a description. That is, to express the set is in words and delimited by curly brackets { } . Example:

  • The T set is the traffic color sets.
  • Association B is an association of Indonesian Language and Literature Education UNNES students who take a specialization in Script Editing.
  • The set Y is the first 7 letters in the order of the Latin alphabet.

Set Types

The existence of a set is not solely one, yes, but there are 4 types that have their own characteristics. Well, here is the explanation.

1. Empty Set

As the name implies, this one set does not have or does not even have any members. The Empty Set will be denoted by a symbol in the form of Φ or { } . In practice, many people cannot distinguish between an empty set and an imprecise set (not a set).

In the empty set, this occurs when the members really do not exist, so that the set or sets are included in the empty set. However, if the members are not clear, in the sense that it cannot be distinguished whether the object in question includes members or not, then the set is not a set, right ? Example of an empty set:

  • Set S is a group of students majoring in English Literature who are 6 years old.
  • The W set is the set of days that start with the letter “H”.
  • The set G is the set of odd numbers that are divisible by 2.

To understand the existence of this empty set, you have to be careful with the number zero (0), right… This is because the number zero (0) is not the empty set, but rather a member of the set which is indeed worth zero (0). For example, there is a set of 5 first whole numbers, so of course the number 0 is a member of that set.

2. The Set of the Universe (Universum)

That is a set that can contain all the objects being discussed. This universal set is also called the universal set of talks, aka the universe set, so it will be denoted as S or U. For example:

  • A collection of Sudirman Kindergarten children wearing white masks

Then the universal set is the set of all Sudirman Kindergarten children.

  • The set of days that start with the letter S.

Then the universal set is the set of the names of the days during the week.

  • B = {red, yellow, green}

Then, the set of possible universes is S = {traffic light colors} or S = {rainbow colors}

3. Finite set

That is a set that has a finite number of members alias can be counted. This type of set is often called a finite set. Example:

  • A = {x│x natural number <7}.

If written in tabular form then A = {1, 2, 3, 4, 5, 6}. The number of finite members of the set A (can be counted), namely 6 (six).

4. Infinite Set

That is the type of set that has infinite members alias cannot be counted, so it is impossible to write in detail, especially when using the tabulation method. So what can be done is to use the sign “…” (three dots) which reads “ next ”. This set is also known as the infinite set. Example:

B = {x│x natural number >15}

Then B can be written as B = {16, 17, 18,…}

Read set B is the set of numbers 16, 17, 18 and so on.

So, that’s an overview of what a set is, along with its history, notation, ways of expressing it, and its types. Has Sinaumed’s been able to make an example of this set based on the objects around you?

Source:

Nugraha, Ali and A.Sy. Dina Dwiyana. MODULE 1: Sets ( http://repository.ut.ac.id/ )

Mahmud, Amir, et al. (2020). COLLECTIONS: Theory and Example Problems . Malang: Expertmedia Press ( http://repository.radenintan.ac.id/ )

Darwanto, et al. (2020). Set Theory . Lampung: Muhammadiyah University Kotabumi. ( http://repository.umko.ac.id/ )

https://evan_ramdan.staff.gunadarma.ac.id/

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