SET THEORY

Set : 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’.

So we can, "A set is a well-defined collection of objects".

The following points may be noted :

(i) Objects, elements and members of a set are synonymous terms.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

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 first five months of a year’ is a set, but ‘the collection of 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. A collection of “Red flowers” is a set, because every red flowers will be included in this set i.e., the objects of the set are well-defined.

2. The collection of past presidents of the United States union is a set.

3. 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.

4. A collection of ‘lovely flowers’ is not a set, because the objects (flowers) to be included are not well-defined.

Reason: The word “lovely” is a relative term. What may appear lovely to one person may not be so to the other person.

5. A collection of “Yellow flowers” is a set, because every red flowers will be included in this set i.e., the objects of the set are well-defined.

Notation:

If a is an element of a set A, we say that “ a belongs to A” the Greek symbol ∈

(epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ∈ A. If ‘b’ is 

not an element of a set A, we write b ∉ A and read “b does not belong to A”.

Thus, in the set V of vowels in the English alphabet, a ∈ V but b ∉ V. In the set

P of prime factors of 30, 3 ∈ P but 15 ∉ P.

There are two methods of representing a set :

(i) Roster or tabular form

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".

Examples:

Set of even numbers: {..., −4, −2, 0, 2, 4, ...}
Set of odd numbers: {..., −3, −1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}