Let \( A = \{2, 3, 6, 7\} \) and \( B = \{4, 5, 6, 8\} \). Let \( R \) be a relation defined on \( A \times B \) by \((a_1, b_1) R (a_2, b_2)\) if and only if \(a_1 + a_2 = b_1 + b_2\). Then the number of elements in \( R \) is __________.
Correct Answer: 25
Approach Solution - 1
Given the sets:
\[ A = \{2, 3, 6, 7\}, \quad B = \{2, 5, 6, 8\} \]
The relation \( (a_1, b_1) \, R \, (a_2, b_2) \) is defined by:
\[ a_1 + a_2 = b_1 + b_2 \]
We list all possible valid pairs \((a_1, b_1)\) and \((a_2, b_2)\) satisfying the condition:
\[ \begin{aligned} 1. &(2, 4)R(6, 4) &\quad 2. &(2, 4)R(7, 5) \\ 3. &(2, 5)R(7, 4) &\quad 4. &(3, 4)R(6, 5) \\ 5. &(3, 5)R(6, 4) &\quad 6. &(3, 5)R(7, 5) \\ 7. &(3, 6)R(7, 4) &\quad 8. &(3, 4)R(7, 6) \\ 9. &(6, 5)R(7, 8) &\quad 10. &(6, 8)R(7, 5) \\ 11. &(7, 8)R(7, 6) &\quad 12. &(6, 8)R(6, 4) \\ 13. &(6, 6)R(6, 6) \end{aligned} \] × 2
Thus, the total number of such relations is:
\[ 24 + 1 = 25 \]
Approach Solution -2
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:
- \((2, 4) \, R \, (6, 4)\): \(2 + 6 = 4 + 4\).
- \((2, 4) \, R \, (7, 5)\): \(2 + 7 = 4 + 5\).
- \((2, 5) \, R \, (7, 4)\): \(2 + 7 = 5 + 4\).
- Similarly, other combinations are checked.
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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