Question

Write the following sets using rule method:
(i) $A = \left\{ {3,6,9,12,15,......} \right\}$
(ii) $P = \left\{ {1,8,27,64,125} \right\}$

Answer 1

Hint: Rule method is also known as set builder form. In this form, each and every element of the set has a single common property which is not possessed by any other element outside the set. In this method, we specify the rule or property or statement. For example: in the set $\left\{ {a,e,i,o,u} \right\}$ all the elements have a common property that each letter is a vowel of English alphabet. This set can be denoted by A and can be written as –
$A = \left\{ {x:x{\text{ is a vowel in english alphabet}}} \right\}$
This is read as “A is a set of all x such that x is a vowel of English alphabet”. Here Sign (:) or (I) is read as “such that”.
In the given problem we follow these steps for writing a given set in set builder form.

Complete step by step solution:
Given sets
(i) $A = \left\{ {3,6,9,12,15,......} \right\}$
(ii) $P = \left\{ {1,8,27,64,125} \right\}$
Since the given sets are written in the roster or tabular form.
Now we have to write these sets from roster or tabular form to set builder form (rule method).
(i) $A = \left\{ {3,6,9,12,15,......} \right\}$
To write in set builder form all the elements of a set should have a common property.
So in the set $\left\{ {3,6,9,12,15,......} \right\}$ all the elements have a common property such that each number is the multiple of 3. So this set can be written in set builder form.
Therefore set builder form of set ‘A’ is
$A = \left\{ {{\text{x : x is the multiple of 3 }}} \right\}$
 or we can also write
$A = \left\{ {{\text{x : x=3n, where n \in N }}} \right\}$

(ii) $P = \left\{ {1,8,27,64,125} \right\}$
In set $\left\{ {1,8,27,64,125} \right\}$ all the elements have a common property such that each number of set is the cube of a natural number. So this set can be written in the set builder form.
Therefore set builder form of set ‘P’ is
$P = \left\{ {x{\text{ : }}x = {n^2}{\text{ }}where{\text{ }}n \in N{\text{ }}and{\text{ }}1 \leqslant n \leqslant 5} \right\}$

Note:
A set can be defined as “a well-defined collection of objects”. Well-defined means whether a particular object belongs to a set or not. A set can be represented by two form roster or tabular form and set builder form (rule method) and a set can be converted from one form to other form. In the given problem we convert the sets from roster or tabular form to set builder form.