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.
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Let $ A $ be the set of all functions $ f: \mathbb{Z} \to \mathbb{Z} $ and $ R $ be a relation on $ A $ such that $$ R = \{ (f, g) : f(0) = g(1) \text{ and } f(1) = g(0) \} $$ Then $ R $ is:
If the domain of the function $ f(x) = \log_7(1 - \log_4(x^2 - 9x + 18)) $ is $ (\alpha, \beta) \cup (\gamma, \delta) $, then $ \alpha + \beta + \gamma + \delta $ is equal to
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