Set, Relations and FunctionsMTP June 22Question 1993 of 217
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Let R be a relation on N defined by x+2y=8\displaystyle x+2y=8. The domain of R is:

Options

A{2,4,8}\displaystyle \{2, 4, 8\}
B{2,4,6,8}\displaystyle \{2, 4, 6, 8\}
C{2,4,6}\displaystyle \{2, 4, 6\}
D{1,2,3,4}\displaystyle \{1, 2, 3, 4\}
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Correct Answer

✅ Option c — {2,4,6}\displaystyle \{2, 4, 6\}

All Options:

  • A{2,4,8}\displaystyle \{2, 4, 8\}
  • B{2,4,6,8}\displaystyle \{2, 4, 6, 8\}
  • C{2,4,6}\displaystyle \{2, 4, 6\}
  • D{1,2,3,4}\displaystyle \{1, 2, 3, 4\}

Detailed Solution & Explanation

A relation R\displaystyle R on the set of natural numbers N={1,2,3,… }\displaystyle \mathbb{N} = \{1, 2, 3, \dots\} is defined by:
x+2y=8x + 2y = 8
We want to find the domain of R\displaystyle R, which is the set of all possible values of x∈N\displaystyle x \in \mathbb{N} that satisfy the relation for some y∈N\displaystyle y \in \mathbb{N}.

Let us solve for y\displaystyle y in terms of x\displaystyle x:
2y=8−x2y = 8 - x
y=8−x2y = \frac{8 - x}{2}
Since y\displaystyle y must be a natural number (y∈N\displaystyle y \in \mathbb{N}), two conditions must be satisfied:
1. y≥1  ⟹  8−x2≥1  ⟹  8−x≥2  ⟹  x≤6\displaystyle y \ge 1 \implies \frac{8 - x}{2} \ge 1 \implies 8 - x \ge 2 \implies x \le 6
2. 8−x\displaystyle 8 - x must be an even integer, which means x\displaystyle x must be an even integer.

Since x\displaystyle x must be a natural number (x∈N\displaystyle x \in \mathbb{N}), the only even integers satisfying 1≤x≤6\displaystyle 1 \le x \le 6 are:
x∈{2,4,6}x \in \{2, 4, 6\}

Let us verify the corresponding values of y\displaystyle y for each x\displaystyle x:
- For x=2\displaystyle x = 2: y=8−22=3∈N\displaystyle y = \frac{8-2}{2} = 3 \in \mathbb{N} (valid pair: (2,3)\displaystyle (2, 3))
- For x=4\displaystyle x = 4: y=8−42=2∈N\displaystyle y = \frac{8-4}{2} = 2 \in \mathbb{N} (valid pair: (4,2)\displaystyle (4, 2))
- For x=6\displaystyle x = 6: y=8−62=1∈N\displaystyle y = \frac{8-6}{2} = 1 \in \mathbb{N} (valid pair: (6,1)\displaystyle (6, 1))

The domain of R\displaystyle R is the set of first components of the relation: {2,4,6}\displaystyle \{2, 4, 6\}.

Hence, **Option C** is the correct answer.

More Questions from Set, Relations and Functions

If A={1,2,3,4,5}\displaystyle A = \{1, 2, 3, 4, 5\}, B={2,4}\displaystyle B = \{2, 4\} and C={1,3,5}\displaystyle C = \{1, 3, 5\} then (A−C)×B\displaystyle (A - C) \times B is:

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