Sets: In Theory and Life

Mathematics deals with different sorts of objects like Numbers (real, integers etc.), lines, triangles, circles, angles, functions and many more. Often, objects with common properties are collected into different groups. Looking at a much bigger picture, there are many things in life which we except are of no use, but we know that “knowledge never goes wasted”. In previous classes we've read much more about 'Set'. Now a question arises in our mind “What is Set?”. The most straightforward definition of a Set is “an unordered collection of objects”. We are familiar with the words tea set, dinner set, sofa set etc. All of these are different but they have common word 'Set'. Now consider dinner set, a collection of different types of items like spoons, bowls, glasses, plates etc., used for serving food. These items are collected and placed in a place because uses of those are same. With the help of this example, we conclude that the collected items having same meaning (common property) formed a set. Here same meaning is “used for serving food”.

   In the development of mathematics, Set theory is very helpful to give proof of many problems which were very complicated in the classical mathematics. Mathematicians are using sets since the very beginning of the subject, for example, Greek mathematicians defined a circle as the set of points which are at a fixed distance 'r' from a fixed-point 'P'. In the late nineteenth century, the mathematician George Cantor (1845–1918) created and developed a mathematical theory of sets and he expressed it as “Collection into a whole M of definite and separate objects of our intuition or our thought. These objects are called the elements of M.”  

  Hence, a set is a well-defined collection of objects and these objects are called elements of the set. Here well-defined means, it must be clear (without any doubt) whether a particular element belongs to the set or not. For example: ‘The collection of natural numbers less than 10’ is a set, because, we can find out whether a given number belongs to this collection or not. But ‘the collection of good students in a class’ is not a set since the term 'good student' is not well-defined. Thus, ‘the collection of first three months of a year’ is a set, but ‘the collection of honest persons in a town’ is not a set. Apart from their mathematical usage, we use sets in our daily life. Some real-life examples of Sets are:

·         In kitchen, utensils/crockery is the most relevant example of sets. In kitchen, plates, bowls, cups etc. are kept separately. Here, plates are considered as one set, bowls and cups are considered as different sets.

·         In a shopping mall, we all have noticed that there are separate sections/floors for each kind of things. For instances, clothing shops are on one section/floor whereas the food court is at another section/floor of the mall.

·         In smartphones/computers, most of us have different kinds of songs playlists like rock, classic and these songs are often separated from each other. Hence, different kind of playlists also form the example of sets.

·         The universe act as a set, as we know that there are millions of galaxies which are separated from each other by some distance.

·         Every company/department/school have different sets of rules such as rules for leave, disciplinary rules and many other, which have to follow by every employee/students. Hence, these different types of rules are considered a sets of rules.

  Of course, the set theory is a very important and exciting branch of mathematics as it offers a foundation with significant applications to other areas of mathematics. The concepts and techniques developed within set theory has attracted many researcher's attention to the philosophy of set theory and the search for new axioms.