Set can be represented in any one of the following three ways or forms.

(i) Descriptive form

(ii) Set-builder form or Rule form

(iii) Roster form or Tabular form

Let us discuss the above different forms representation of a set in detail.

One way to specify a set is to give a verbal description of its elements.

This is known as the descriptive form of specification.

The description must allow a concise determination of which elements belong to the set and which elements do not.

For example,

(i) The set of all natural numbers.

(ii) The set of all prime numbers less than 100.

(iii) The set of all letters in English alphabets.

Set-builder notation is a notation for describing a set by indicating the properties that its members must satisfy.

**Reading Notation :**

A = { x : x is a letter in the word "dictionary" }

We read it as

“A is the set of all x such that x is a letter in the word dictionary”

For example,

(i) N = { x : x is a natural number }

(ii) P = { x : x is a prime number less than 100 }

(iii) A = { x : x is a letter in the English alphabet }

Listing the elements of a set inside a pair of braces { } is called the roster form.

For example,

(i) Let A be the set of even natural numbers less than 11.

In roster form we write,

A = {2, 4, 6, 8, 10}

(ii) A = {x : x is an integer and -1 ≤ x < 5}

In roster form we write,

A = {-1, 0,1, 2, 3, 4}

(i) In roster form each element of the set must be listed exactly once. By convention, the elements in a set should not be repeated.

(ii) Let A be the set of letters in the word “coffee”,

That is, A = {c, o, f, e}. So, in roster form of the set A the following are invalid.

{c, o, e} -------> (not all elements are listed)

{c, o, f, f, e} -------> (element ‘f’ is listed twice)

(iii) In a roster form the elements in a set can be written in any order.

The following are valid roster form of the set containing the elements 2, 3 and 4.

{2, 3, 4}

{2, 4, 3}

{4, 3, 2}

Each of them represents the same set.

(iv) If there are either infinitely many elements or a large finite number of elements, then three consecutive dots called ellipsis are used to indicate that the pattern of the listed elements continues, as in

{5, 6, 7,......} or {3, 6, 9, 12, 15,........60}

(v) Ellipsis can be used only if enough information has been given so that one can figure out the entire pattern.

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