Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A × B such that R = {(\((a_1, b_1),(a_2, b_2)\)) : \(a_1 ≤ b_2\) and \(b_1 ≤ a_2\)}. Then the number of elements in the set R is
- 180
- 26
- 160
- 52
The Correct Option is C
Solution and Explanation
Step 1: Analyze the given conditions for \(a_1\) and \(b_2\)
We are given the following choices:
- \(a_1 = 1 \Rightarrow 5\) choices of \(b_2\)
- \(a_1 = 3 \Rightarrow 4\) choices of \(b_2\)
- \(a_1 = 4 \Rightarrow 4\) choices of \(b_2\)
- \(a_1 = 6 \Rightarrow 2\) choices of \(b_2\)
- \(a_1 = 9 \Rightarrow 1\) choice of \(b_2\)
Total number of ways for \((a_1, b_2)\):
5 + 4 + 4 + 2 + 1 = 16 ways.
Step 2: Analyze choices for \(b_1\) and \(a_2\)
We are given the following choices:
- \(b_1 = 2 \Rightarrow 4\) choices of \(a_2\)
- \(b_1 = 4 \Rightarrow 3\) choices of \(a_2\)
- \(b_1 = 5 \Rightarrow 2\) choices of \(a_2\)
- \(b_1 = 8 \Rightarrow 1\) choice of \(a_2\)
Total number of ways for \((b_1, a_2)\):
4 + 3 + 2 + 1 = 10 ways.
Step 3: Calculate the total number of required elements
Combining the above results, the total number of required elements in \(R\) is:
\[ \text{Total required elements in } R = 16 \times 10 = 160 \]
Final Answer
The required number of elements in \(R\) is 160.
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