Question

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

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

Step-by-step solution

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.

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).

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}

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.

Key concepts

Natural Numbers

Natural numbers are the foundation of mathematics. They start from 1 and go upwards infinitely like 1, 2, 3, and so on. These numbers do not include zero or any fractional numbers. They are here basically to count and order items. It's like saying, "I have 3 apples and 4 oranges." Natural numbers make sure you count your things in whole units.
Understanding natural numbers is crucial when working with equations and relations. They give way to ensure that all values being talked about are whole and positive. This prevents the results from falling into fractions or becoming negative, which we might not want in specific scenarios such as this exercise.

Equation Solving

The art of equation solving is about finding an unknown value that makes an equation true. In our exercise, we have the equation: \(x + 2y = 8\).This equation presents a relationship between two variables: \(x\) and \(y\). To solve it with natural numbers, the first step is to assume possible values for one variable and then find the corresponding value of the other variable.

• Start with as low a value as possible for one variable, like choosing \(y = 1\).
• Substitute forward. When \(y = 1\), \(x\) becomes \(8 - 2(1)\), which equals 6.
• Continue checking other values for \(y\) and repeat the substitution until you find viable solutions.

Each pair must satisfy the given equation while both \(x\) and \(y\) remain natural numbers. Through this process, valid pairs can be found, as illustrated in the solution.

Domain of a Relation

The domain of a relation in mathematics refers to the set of all first elements from the ordered pairs that form that relation. In simpler terms, it's the list of all possible values that \(x\) can take, in relation \(R\). Given a relation \(R = \{(x, y) \mid x + 2y = 8\}\),we find the domain by determining all the possible \(x\) values that satisfy the equation with natural numbers.

• By solving \(x + 2y = 8\), and iterating through different values for \(y\), one can find every possible \(x\) that results in a natural number.
• As observed, at \(y = 1\), \(x = 6\); \(y = 2\), \(x = 4\); and \(y = 3\), \(x = 2\). These are our \(x\) values for the domain.
• Finding or listing these values forms the domain: \(\{6, 4, 2\}\).

Thus, ensuring that the domain includes only natural numbers avoids any unrealistic solutions in this context.