Sets

Set theory is about studying collection of objects. The collection may comprise anything or any abstraction. It can be purely abstract thing like numbers or abstraction of real thing like students studying in class XI in a school. The members of collection can be numbers, letters, titles of books, people, teachers, provinces – virtually anything - even other collections. Further, it need not be finite. For example, a set of integers has infinite members. For a set, only requirement is that the members of a collection are properly defined.

Set

A set is a collection of well defined objects.

In other words, the member of set is clearly identifiable. The terms “object”, “member” or “element” mean same thing and are used interchangeably.

How to represent a set?

A set is denoted by capital letters like “A”, “B”, “C” etc. In choosing a symbol for a set, it is generally convenient to use a capital letter that identifies with the set. For example, it is appropriate to use symbol “V” to represent collection of vowels in English alphabet.

On the other hand, the members or elements of a set are denoted by small letters like “a”,”b”,”c” etc.

Membership of a set is denoted by the symbol “ $\in$ ” . Its literal meaning is “belongs to”. If an object does not belong to a set, then we convey the same, using symbol “ $\notin$ ”.

$a\in A$ : we read this as “a” belongs to set "A".

$a\notin A$ : we read this as “a” does not belong to set "A".

The set is represented in two ways :

• Roaster form
• Set builder form

Roaster form

All elements of the set are listed with a comma (“,”) in between and the listing itself is enclosed within braces “{“ and “}”. The order or sequence of elements within the set is not important – though desirable.

The set of numbers, which divide 12, is written as :

$$A=\left\{\mathrm{1,2,3,4,6,12}\right\}$$

If a pattern or sequence is easily made out, then we can use ellipsis ("...") to represent continuity of such pattern. This type of representation is particularly useful to represent an infinite set. Clearly, sequence of members in this type of representation is important.

The set of even numbers is written as,

$$B=\{\mathrm{2,4,6,8}\dots \dots \dots \}$$

The roaster form is limited in certain circumstance. For example, we can not represent set of real numbers in roaster form. Real numbers is an infinite set, but the elements of this set do not follow a pattern or have a particular sequence. As such, we can not define same with the help of ellipsis.

Every member of the set is unique and distinct. However, we encounter situations in which collection can have repeated elements. For example, the set representing scores of three students can be a set of three numbers one of which is repeated :

$$S=\left\{\mathrm{80,80,70}\right\}$$

We need to reduce such collection as :

$$\Rightarrow S=\left\{\mathrm{80,80,70}\right\}=\left\{\mathrm{80,70}\right\}$$

Set builder form

Collections are often characterized by a common property. We can, therefore, define members of a set in terms of the common property. However, we need to ensure that objects outside the collection do not have the same common property.

The construction of qualification for the common property is quite flexible. Only thing is that we need to be explicit in what we mean. Generally, we denote the member by a symbol like “x” and then define the membership. Consider the examples :