Question

The relation $R$ defined on the set of natural numbers as $\{ \left( {a,b} \right):a$ differs from $b$ by $3\} $ is given by
1. $\left\{ {\left( {1,4} \right),\left( {2,5} \right),\left( {3,6} \right),......} \right\}$
2. $\left\{ {\left( {4,1} \right),\left( {5,2} \right),\left( {6,3} \right),......} \right\}$
3. $\left\{ {\left( {1,3} \right),\left( {2,61} \right),\left( {3,9} \right),......} \right\}$
4. None of the above

Answer 1

Hint: In the given problem, we are given the relation $R$ which is defined on the set of natural numbers. Using the condition $a$ differs from $b$ by $3$ , make the equation then find the value of $a$ and write $a,b$ as a single constant $n$(any). Put $n = 1,2,3,.........$ for the required relation.

Complete step by step solution: 
Given that,
$\{ \left( {a,b} \right):a$ differs from $b$ by $3\} $
$ \Rightarrow a - 3 = b$
$a = b + 3$
Here, $R = \left\{ {\left( {a,b} \right):a,b \in N,a - b = 3} \right\}$
$R = \left\{ {\left( {n + 3,n} \right):n \in N,b = n} \right\}$
Where $n = 1,2,3,.........$
$R = \left\{ {\left( {4,1} \right),\left( {5,2} \right),\left( {6,3} \right),......} \right\}$
Therefore, the correct option is 2.

Note: The key concept involved in solving this problem is the good knowledge of sets, relations, and functions. Students must remember that Sets are collections of ordered elements, whereas relations and functions are actions on sets. The relations establish the link between the two specified sets. There are also other sorts of relations that describe the links between the sets.