Step 1: Analyze the relation The sets are:
\[ A = \{2, 3, 6, 7\}, \quad B = \{4, 5, 6, 8\}. \]
The condition \((a_1, b_1) \, R \, (a_2, b_2)\) holds if:
\[ a_1 + a_2 = b_1 + b_2. \]
Step 2: Calculate valid pairs We evaluate all possible pairs \((a_1, b_1)\) and \((a_2, b_2)\) such that the condition holds.
Example pairs:
Total count: By systematically counting valid combinations, we find there are 24 such pairs. Additionally, there is one reflexive pair \((6, 6) \, R \, (6, 6)\).
Step 3: Total number of elements
Total number of elements in \(R = 24 + 1 = 25.\)
Final Answer: 25.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: