Question

Let \( A = \{2,3,4,5,6\} \) be a set. Consider \( R \) be a relation on \( A \times A \) such that \( (x,y) \, R \, (a,b) \) implies \( x \) divides \( a \) and \( y \leq b \), then total number of elements in \( R \) is:

• A. 24
• B. 120
• C. 720
• D. 144

Hint:

When a relation on \(A \times A\) is defined by two independent conditions, count each condition separately and multiply the totals. This makes such counting problems much faster.

Answer 1

The Correct Option is B

Step 1: Understand what has to be counted.
The relation \( R \) is defined on \( A \times A \). So, an element of \( R \) is of the form
\[ ((x,y),(a,b)) \] such that:
\[ x \text{ divides } a \quad \text{and} \quad y \leq b \] Hence, for each fixed pair \( (a,b) \in A \times A \), we count the number of possible \( x \) and \( y \) satisfying these conditions.

Step 2: Count values of \(x\) such that \(x\mid a\).
Now \( A=\{2,3,4,5,6\} \). For each \( a \in A \), count how many elements of \( A \) divide \( a \):
\[ a=2 \Rightarrow \{2\} \Rightarrow 1 \] \[ a=3 \Rightarrow \{3\} \Rightarrow 1 \] \[ a=4 \Rightarrow \{2,4\} \Rightarrow 2 \] \[ a=5 \Rightarrow \{5\} \Rightarrow 1 \] \[ a=6 \Rightarrow \{2,3,6\} \Rightarrow 3 \] So, total divisor-count contribution is:
\[ 1+1+2+1+3=8 \]
Step 3: Count values of \(y\) such that \(y \leq b\).
Again \( A=\{2,3,4,5,6\} \). For each \( b \in A \), count how many elements of \( A \) are less than or equal to \( b \):
\[ b=2 \Rightarrow \{2\} \Rightarrow 1 \] \[ b=3 \Rightarrow \{2,3\} \Rightarrow 2 \] \[ b=4 \Rightarrow \{2,3,4\} \Rightarrow 3 \] \[ b=5 \Rightarrow \{2,3,4,5\} \Rightarrow 4 \] \[ b=6 \Rightarrow \{2,3,4,5,6\} \Rightarrow 5 \] So, total \(y\)-count contribution is:
\[ 1+2+3+4+5=15 \]
Step 4: Find the total number of elements in \(R\).
For each \( (a,b) \), the number of valid pairs \( (x,y) \) is:
\[ (\text{number of }x\text{ dividing }a)\times(\text{number of }y\leq b) \] Therefore, total number of elements in \( R \) is:
\[ \left( \sum \text{divisor counts} \right)\left( \sum \text{order counts} \right) \] \[ =8 \times 15 \] \[ =120 \]
Step 5: Conclusion.
Hence, the total number of elements in the relation \( R \) is \(120\).
Final Answer: \(120\)