Question

Define a relation R on the set N of natural numbers by $R=\left\{ \left( x,y \right):y=x+5, \right.$ x is a natural number less than $\left. \text{4;x,y}\in \text{N} \right\}$. Depict this relationship using Roster form. Write down the domain and range.

Answer 1

Hint: We are asked to depict the relation $R=\left\{ \left( x,y \right):y=x+5, \right.$ x is a natural number less than $\left. \text{4;x,y}\in \text{N} \right\}$ into Roster form. So, we will first find the possible values of x. As it is a natural number, it would start from 1 and be less than 4 as per the condition. So, possible values of x are 1, 2 and 3. Then using y=x+5, we will find the value of y by substituting x as 1, 2 and 3. So, using this we will get the Roster form. Then, we will use the fact that the domain of relation consists of the first element of R while range consists of all second elements of R to find domain and range.

Complete step by step answer:
We are given that, a relation R on the set N is defined as
$R=\left\{ \left( x,y \right):y=x+5, \right.$ x is a natural number less than $\left. \text{4;x,y}\in \text{N} \right\}$
To define the relation R, we will first find the possible values of x.
We are given that x is a natural number less than 4. We know natural numbers start from 1 and go on as 1, 2, 3, 4 . . . . . . . .
So, natural numbers less than 4 are 1, 2 and 3.
So, we get the possible values of x as 1, 2 and 3.
Now, R is defined as \[R=\left\{ \left( x,y \right):y=x+5,x=1,2,3 \right\}\]
Now, when x = 1, we have
\[\Rightarrow y=x+5=1+5=6\]
So, for x = 1 we have y = 6. So, we have first pair as \[\left( 1,6 \right)\in R\]
When x = 2, we have
\[\Rightarrow y=x+5=2+5=7\]
So, for x = 2 we have y = 7. Now, second pair is \[\left( 2,7 \right)\in R\]
And lastly for x = 3, we have
\[\Rightarrow y=x+5=3+5=8\]
So, for x = 3 we have y = 8. So, we have got the last pair as \[\left( 3,8 \right)\in R\]
We know Roster form is a representation of a set that lists all the elements of the set.
Our relation R has elements as follows: \[\left( \text{1},\text{ 6} \right),\text{ }\left( \text{2},\text{ 7} \right)\text{ and }\left( \text{3},\text{ 8} \right)\].
So, the Roster form of R that we get is
\[R=\left\{ \left( \text{1},\text{ 6} \right),\left( \text{2},\text{ 7} \right),\left( \text{3},\text{ 8} \right) \right\}\]
Now, we know that the domain of R is defined as the set of all first elements in relation. We have possible values of x as 1, 2 and 3 only so, we get
\[\text{Domain of R}=\left\{ \text{1},\text{2},\text{3} \right\}\]
We also know that the Range of R is defined as the set of all second elements in the relation. We have possible values of y as 6, 7 and 8 only so, we get
\[\text{Range of R}=\left\{ \text{6},\text{7},\text{8} \right\}\]

Note:
Remember that the restriction of numbers being less than 4 is only on x and not on y. Natural numbers do not include 0. To find the Roster form we find all elements that belong to R and then we will consider those as a set and that set will become a Roster form of R.