Question

Let \(R\) be a relation on natural numbers defined by \(x+2y=8,\ x,y\in N\). The domain of \(R\) is

• A. \(\{2,4,6,8\}\)
• B. \(\{2,4,6\}\)
• C. \(\{2,4,8\}\)
• D. \(\{1,2,3\}\)

Hint:

The domain of a relation is the set of all first components \(x\) of ordered pairs satisfying the given relation.

Answer 1

The Correct Option is B

Step 1: Understand the relation.
The relation is given by:
\[ x+2y=8, \] where
\[ x,y\in N. \]

Step 2: Express \(x\) in terms of \(y\).
\[ x=8-2y. \]

Step 3: Use natural number values of \(y\).
Since \(y\in N\), take positive natural values:
\[ y=1,2,3,\ldots \]

Step 4: Find corresponding values of \(x\).
For \(y=1\):
\[ x=8-2(1)=6. \]
For \(y=2\):
\[ x=8-2(2)=4. \]
For \(y=3\):
\[ x=8-2(3)=2. \]

Step 5: Check next value.
For \(y=4\):
\[ x=8-2(4)=0. \]
But \(0\) is not included in natural numbers here.

Step 6: Write valid \(x\)-values.
Valid values of \(x\) are:
\[ 6,\ 4,\ 2. \]
So, the domain is:
\[ \{2,4,6\}. \]

Step 7: Final conclusion.
Thus, the domain of relation \(R\) is:
\[ \boxed{\{2,4,6\}}. \]