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Chapter 1: Problem 49

Let \(R\) be a relation on \(\mathrm{N}\) defined by \(\mathrm{R}=\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}+2 \mathrm{y}=8\\}\) The domain of \(\mathrm{R}\) is (a) \(\\{2,4,8\\}\) (b) \(\\{2,4,6,8\\}\) (c) \(\\{2,4,6\\}\) (d) \(\\{1,2,3,4\\}\)

Short Answer

Expert verified
The domain of the relation R is (c) $\{2, 4, 6\}$.

Step by step solution

01

Analyzing the given relation

To determine the possible x values for which given relation holds true, we need to analyze x + 2y = 8 and determine its constraints within natural numbers N.
02

Determine possible x values

Since both x and y are natural numbers, we know that 0 < x and 0 < y. So, we need to look for pairs of (x, y) in N for which x + 2y = 8. By looking at the given equation, we see that x = 8 - 2y, which means that when y is the smallest possible natural number (1), x should be the maximum possible value. So when y = 1: x = 8 - 2(1) = 6 Now, let's find out other possible values of y for which the value of x remains a natural number. When y = 2: x = 8 - 2(2) = 8 - 4 = 4 When y = 3: x = 8 - 2(3) = 8 - 6 = 2 Notice that when y = 4: x = 8 - 2(4) = 8 - 8 = 0, but x cannot be 0 because it is a natural number, so the possibilities of (x, y) are (6, 1), (4, 2) and (2, 3).
03

Determine the domain of relation R

The domain of relation R consists of all the x values in pairs (x, y) for which x + 2y = 8, which we found in Step 2. So, the domain of R = {x for the pairs (x, y) = (6, 1), (4, 2), (2, 3)} = {2, 4, 6}
04

Select the correct answer

Now, comparing our domain of R with the given options, we can see that it matches with option (c): {2, 4, 6}. Therefore, option (c) is the correct answer.

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