Let R be a relation from A = \[\left\{ {1,2,3,4} \right\}\]to B = \[\left\{ {1,3,5} \right\}\]i.e. \[\left( {a,b} \right) \in R\], if \[a{\rm{ }} < {\rm{ }}b\] then find \[RO{R^{ - 1}}\]

Hint: According the question find out the relation R from A and B if \[a{\rm{ }} < {\rm{ }}b\] and also find out inverse using relation R. Then calculate \[RO{R^{ - 1}}\].

Complete step-by-step answer:
Firstly, here we will calculate the relation R that is \[(a,b)\] that is to be formed by using the condition \[a{\rm{ }} < {\rm{ }}b\] .
It is given that A = \[\left\{ {1,2,3,4} \right\}\]and B = \[\left\{ {1,3,5} \right\}\].
So, relation R = \[\left\{ {\left( {1,3} \right),\left( {1,5} \right),\left( {2,3} \right),\left( {2,5} \right),\left( {3,5} \right),\left( {4,5} \right)} \right\}\]
Now, we will calculate \[{R^{ - 1}}\] that is \[(b,a)\] by reversing all the set values in relation R.
So, \[{R^{ - 1}} = \left\{ {\left( {3,1} \right),\left( {5,1} \right),\left( {3,2} \right),\left( {5,2} \right),\left( {3,5} \right),\left( {5,4} \right)} \right\}\]
Here, taking one by one all the values of relation \[{R^{ - 1}}\] that is \[(a,b)\]and then find out in relation R which is starting from b that is \[(b,c)\] . Through which we can calculate the relation \[RO{R^{ - 1}}\]that is \[(a,c)\] .
As, \[RO{R^{ - 1}} = \left( {3,1} \right) \in {R^{ - 1}}\] and \[\left( {1,5} \right) \in R\]
Then, \[\left( {3,5} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {3,1} \right) \in {R^{ - 1}}\] and \[\left( {1,3} \right) \in R\]
Then, \[\left( {3,3} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {5,1} \right) \in {R^{ - 1}}\] and \[\left( {1,3} \right) \in R\]
Then, \[\left( {5,3} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {5,1} \right) \in {R^{ - 1}}\] and \[\left( {1,5} \right) \in R\]
Then, \[\left( {5,5} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {3,2} \right) \in {R^{ - 1}}\] and \[\left( {2,3} \right) \in R\]
Then, \[\left( {3,3} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {3,2} \right) \in {R^{ - 1}}\] and \[\left( {2,5} \right) \in R\]
Then, \[\left( {3,5} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {5,2} \right) \in {R^{ - 1}}\] and \[\left( {2,3} \right) \in R\]
Then, \[\left( {5,3} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {5,2} \right) \in {R^{ - 1}}\] and \[\left( {2,5} \right) \in R\]
Then, \[\left( {5,5} \right) \in RO{R^{ - 1}}\]
As, \[RO{R^{ - 1}} = \left( {3,5} \right) \in {R^{ - 1}}\]but there is not any set that starts from 5 in relation R. So, \[RO{R^{ - 1}}\] cannot be formed.
As, \[RO{R^{ - 1}} = \left( {5,4} \right) \in {R^{ - 1}}\] and \[\left( {4,5} \right) \in R\]
Then, \[\left( {5,5} \right) \in RO{R^{ - 1}}\]
Therefore as of now, we will take all the values of \[RO{R^{ - 1}}\]without repeating and put them in a relation function.
Hence, \[RO{R^{ - 1}} = \left\{ {\left( {3,3} \right),\left( {3,5} \right),\left( {5,3} \right),\left( {5,5} \right)} \right\}\]

Note: To solve these types of questions, you need to calculate relation R using the given condition. As, in the above question it is required to calculate \[RO{R^{ - 1}}\] from which we also need to calculate \[{R^{ - 1}}\] .
As, it important to see first the value of \[{R^{ - 1}}\] that is \[\left( {a,b} \right)\] then use the values from R that is \[\left( {b,c} \right)\]
And hence \[RO{R^{ - 1}}\]is calculated \[\left( {a,c} \right)\]. So, by following the above method we can calculate any required value.