Idea of Sets

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Welcome to class! 

In today’s class, we will be talking about the idea of sets. Enjoy the class!

Idea of Sets

Basic concepts of sets

To know the basic concepts of sets let us understand from our day to day life we often speak or hear about different types of collections.

Such as:

1. a collection of pens
2. a collection of dolls
3. a collection of books, etc.

In the same way, we have different types of groups made for a different activity such as:

1. a group of boys playing cricket
2. a group of girls playing tennis
3. a group of friends going for a movie, etc.

In mathematics, a collection of particular things or group of particular objects is called a set. The theory of sets as developed George Cantor is being used in all branches of mathematics nowadays. According to him ‘A set is a well-defined collection of distinct objects of our perception or of our thought, to be conceived as a whole’.

As in the case of the concepts of geometrical point, line and a plane, a rigid definition is not possible for a set also. Is the intuitive conception of a collection or assemblage of things, real or conceptual.

The examples of the basic concepts of sets are:

1. a set of living cricketers in Australia.
2. a set of the rules for the badminton game.
3. a set of integers with prescribed conditions.
4. a set of books in the library.
5. a set of the states in America.

Thus, the basic concepts of sets is a well-defined collection of objects which are called members of the set or elements of the set. Objects that belong to the set must be well-distinguished.

Definition of set

A set is a collection of well-defined objects.

Explanation of the term “Well-defined”:

Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.

For example

‘The collection of positive numbers less than 10’ is a set, because, given any numbers, we can always find out whether that number belongs to the collection or not. But ‘the collection of good students in your class’ is not a set as in this case no definite rule is supplied by the help of which you can determine whether a particular student of your class is good or not. Thus, ‘the collection of the first five months of a year’ is a set, but ‘the collection of a rich man in your town’ is not a set.

Now, to get basic concepts of sets about the meaning of well-defined the following examples are given below.

1. The collection of vowels in English alphabets. This set contains five elements, namely, a, e, i, o, u.
2. A group of “Singers with ages between 18 years and 25 years” is a set, because the range of ages of the singer is given and so it can easily be decided that which singer is to be included and which is to be excluded. Hence, the objects are well-defined.
3. A collection of “Red flowers” is a set, because every red flower will be included in this set i.e., the objects of the set are well-defined.
4. The collection of past presidents of the United States union is a set.
5. A group of “Young dancers” is not a set, as the range of the ages of young dancers is not given and so it can’t be decided that which dancer is to be considered young i.e., the objects are not well-defined.
6. The collection of cricketers in the world who were out for 99 runs in a test match is a set.

 

In our next class, we will be talking about the Complement of Set.  We hope you enjoyed the class.

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