Question

Let \(R\) be a relation in \(N\) defined by \(R=\{(x,y): x+2y=8\\). The range of \(R\) is}

• A. \(\{2,4,6\}\)
• B. \(\{1,2,3\}\)
• C. \(\{1,2,3,4,6\}\)
• D. None of these

Hint:

Range of a relation means the set of all possible second elements \(y\) in the ordered pairs.

Answer 1

The Correct Option is B

Concept:
A relation \(R\) contains ordered pairs \((x,y)\). The range of a relation is the set of all second components \(y\) of the ordered pairs. Here: \[ R=\{(x,y):x+2y=8\} \] where \(x,y\in N\).

Step 1: Use the given equation.
The relation is defined by: \[ x+2y=8 \] We need possible natural number values of \(y\).

Step 2: Find possible values of \(y\).
If: \[ y=1 \] then: \[ x+2(1)=8 \] \[ x=6 \] So \((6,1)\) is possible. If: \[ y=2 \] then: \[ x+4=8 \] \[ x=4 \] So \((4,2)\) is possible. If: \[ y=3 \] then: \[ x+6=8 \] \[ x=2 \] So \((2,3)\) is possible.

Step 3: Check next value.
If: \[ y=4 \] then: \[ x+8=8 \] \[ x=0 \] But \(0\) is not considered a natural number here. So \(y=4\) is not included.

Step 4: Write the range.
The possible values of \(y\) are: \[ 1,\ 2,\ 3 \] Therefore: \[ \text{Range}=\{1,2,3\} \] Hence, the correct answer is: \[ \boxed{(B)\ \{1,2,3\}} \]