Finding the Number of Elements in a Relation

The problem asks us to find the number of elements in a relation R. The relation R is defined by the equation \(2x + 3y = 20\), and the elements \((x, y)\) must satisfy this equation. Furthermore, the condition given is that both \(x\) and \(y\) belong to the set of natural numbers, denoted by \(\mathbb{N}\). In standard mathematical context, natural numbers \(\mathbb{N}\) are the positive integers starting from 1, i.e., \(\{1, 2, 3, \dots\}\).

Determining Ordered Pairs for the Relation

We need to find pairs of natural numbers \((x, y)\) that satisfy the equation \(2x + 3y = 20\). We can systematically look for such pairs. Since \(x\) and \(y\) are natural numbers, they must be positive integers.

Let's express \(y\) in terms of \(x\) from the equation:

\[3y = 20 - 2x\]

\[y = \frac{20 - 2x}{3}\]

For \(y\) to be a natural number, it must be a positive integer. This requires two conditions:

1. The numerator \(20 - 2x\) must be divisible by 3.
2. The value of \(y\) must be positive, which means \(\frac{20 - 2x}{3} > 0\). This implies \(20 - 2x > 0\), so \(20 > 2x\), or \(10 > x\).

Since \(x\) must be a natural number (\(x \ge 1\)) and \(x < 10\), the possible natural number values for \(x\) are \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

Now, let's test each of these possible \(x\) values to see if \(20 - 2x\) is divisible by 3 and if the resulting \(y\) is positive.

Alternatively, we could start by listing possible values for \(y\). Since \(y\) is a natural number, \(y \ge 1\). Also, \(3y = 20 - 2x\). Since \(x \ge 1\), \(2x \ge 2\), so \(20 - 2x \le 18\). This means \(3y \le 18\), so \(y \le 6\). The possible natural number values for \(y\) are \(\{1, 2, 3, 4, 5, 6\}\).

Let's express \(x\) in terms of \(y\) from the equation:

\[2x = 20 - 3y\]

\[x = \frac{20 - 3y}{2}\]

For \(x\) to be a natural number, it must be a positive integer. This requires:

1. The numerator \(20 - 3y\) must be divisible by 2 (i.e., \(20 - 3y\) must be an even number).
2. The value of \(x\) must be positive, which means \(\frac{20 - 3y}{2} > 0\). This implies \(20 - 3y > 0\), so \(20 > 3y\), or \(y < 20/3 \approx 6.67\).

Since \(y\) must be a natural number (\(y \ge 1\)) and \(y < 6.67\), the possible natural number values for \(y\) are \(\{1, 2, 3, 4, 5, 6\}\).

Now, let's test each of these possible \(y\) values to see if \(20 - 3y\) is even and if the resulting \(x\) is positive.

Both methods yield the same set of ordered pairs \((x, y)\) where both \(x\) and \(y\) are natural numbers satisfying \(2x + 3y = 20\).

The pairs are (1, 6), (4, 4), and (7, 2).

Conclusion: Counting the Relation Elements

The relation R consists of the set of ordered pairs \(\{(1, 6), (4, 4), (7, 2)\}\). To find the number of elements in R, we simply count the number of pairs in this set.

There are 3 distinct ordered pairs in the set.

Therefore, the number of elements in the relation R is 3.

Revision Table: Relation and Natural Numbers

Additional Information: Linear Diophantine Equations and Natural Numbers

The equation \(2x + 3y = 20\) is a type of linear Diophantine equation. These are equations of the form \(ax + by = c\) where integer solutions are sought. Our problem adds the constraint that the solutions must be natural numbers (positive integers).

For a linear Diophantine equation \(ax + by = c\) to have integer solutions, the greatest common divisor (GCD) of \(a\) and \(b\) must divide \(c\). In \(2x + 3y = 20\), \(a=2\), \(b=3\), and \(c=20\). GCD(2, 3) = 1, and 1 divides 20, so integer solutions exist.

Finding natural number solutions often involves finding the general integer solution and then applying the conditions \(x > 0\) and \(y > 0\). As shown in the step-by-step solution by checking values, the possible natural number pairs for this specific equation are limited because the coefficients (2 and 3) and the constant term (20) restrict the range of possible values for \(x\) and \(y\).