Question

Let $R$ be a relation on $N$ defined by$R=\left\{(x,y):x+2y=8\right\}$. The domain of $R$ is

• A. $\left\{2,4,8\right\}$
• B. $\left\{2,4,6,8\right\}$
• C. $\left\{2,4,6\right\}$
• D. $\left\{1,2,3,4\right\}$

Answer 1

The correct option is C

$\left\{2,4,6\right\}$

Explanation For The Correct Options:

Finding the domain of $R$

The given relation

$R=\left\{(x,y):x+2y=8\right\}$

Put $x=2$

$$\begin{gathered} \Rightarrow 2+2y=8 \\ \Rightarrow 2y=6 \\ \Rightarrow y=3\in N \end{gathered}$$

Similarly,

For $x=4$ we get $y=2\in N$,

For $x=6$ we get $y=1\in N$,

Thus, $\left\{2,4,6\right\}$ is the domain of function.

Hence option C is correct.

Explanation for incorrect options:

Option(A):

The given relation

$R=\left\{(x,y):x+2y=8\right\}$

Put $x=8\in$$\left\{2,4,8\right\}$

$$\begin{gathered} \Rightarrow 8+2y=8 \\ \Rightarrow 2y=8 \\ \Rightarrow y=0\notin N \end{gathered}$$

Hence option A is incorrect.

Option(B):

The given relation

$R=\left\{(x,y):x+2y=8\right\}$

Put $x=8\in$$\left\{2,4,6,8\right\}$

$$\begin{gathered} \Rightarrow 8+2y=8 \\ \Rightarrow 2y=8 \\ \Rightarrow y=0\notin N \end{gathered}$$

Hence option (B) is incorrect.

Option(D):

The given relation

$R=\left\{(x,y):x+2y=8\right\}$

Put $x=3\in$$\left\{1,2,3,4\right\}$

$$\begin{gathered} \Rightarrow 3+2y=8 \\ \Rightarrow 2y=5 \\ \Rightarrow y=\frac{5}{2}\notin N \end{gathered}$$

Hence option (D) is incorrect.

Hence, option (C) is the correct answer.