Question

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 

• A. 180
• B. 26
• C. 160
• D. 52

Answer 1

The Correct Option is C

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.