Question

# Let R be a relation on N defined by x+2y=8.The domain of R isA. {2,4,6}B. {2,4,6,8}C. {2,4,6}D. {1,2,3,4}

Hint-Find out the equation for either x or y and then find out the respective values of x and y.

We have been given with a relation R on set of natural numbers defined by x+2y=8
From this, we can write y=$\dfrac{{8 - x}}{2}$
Since, x and y are natural numbers , the value of x and y has to be positive and also zero is excluded,
Also, from the equation , we can infer that the value of x has to be an even number , only then we will get y as a natural number
So, let’s start from the least natural even number and proceed
So, in the first case, we get
If x=2, y=$\dfrac{{8 - 2}}{2} = 3$
If x=4,$y = \dfrac{{8 - 4}}{2} = 2$
If x=6,$y = \dfrac{{8 - 6}}{2} = 1$
If x=8,$y = \dfrac{{8 - 8}}{2} = 0$ , but 0 does not belong to the set of natural numbers
So even the only possible values of x are 2,4,6
So, the relation will be R={(2,3),(4,2),(6,1)}
So, now we can write the domain of the relation R={2,4,6}
So, option C is the correct answer for this question.

Note: Please make sure to write the proper values of the domain when it is asked, do not get confused with the values of the co-domain and whenever we have problems like these make sure to write the values of the domain in the form of a set.