Question

Suppose that \( A = \{2, 3, 4, 5\} \) and a relation \( R \) on \( A \) is defined by \( R = \{(a, b) : a, b \in A, a - b = 12\} \). Then the set \( R \) is

• A. \(\emptyset\)
• B. Not \(\emptyset\)
• C. \{2, 3\}
• D. \{2, 4, 5\}

Hint:

When dealing with relations on finite sets with a condition involving inequalities or specific values, always check the extreme cases first (maximum and minimum possible values). This can often lead to a quick conclusion without checking every single pair.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A relation \( R \) on a set \( A \) is a subset of the Cartesian product \( A \times A \).
The problem asks us to find the elements of the relation \( R \), which are ordered pairs \( (a, b) \) where both \( a \) and \( b \) belong to the set \( A \), and they must satisfy the condition \( a - b = 12 \).
Step 2: Key Formula or Approach:
The approach is to check all possible pairs \( (a, b) \) from \( A \times A \) to see if they satisfy the condition \( a - b = 12 \).
A more efficient approach is to find the maximum possible value of \( a - b \) for the elements in set \( A \).
Step 3: Detailed Explanation:
The given set is \( A = \{2, 3, 4, 5\} \).
The condition for an ordered pair \( (a, b) \) to be in the relation \( R \) is \( a - b = 12 \).
To find the maximum possible value of the difference \( a - b \), we choose the largest possible value for \( a \) and the smallest possible value for \( b \) from the set \( A \).
\[ \text{Maximum value of } a \text{ from } A \text{ is } 5 \] \[ \text{Minimum value of } b \text{ from } A \text{ is } 2 \] Now, calculate the maximum possible difference: \[ \text{Max}(a - b) = \text{Max}(a) - \text{Min}(b) = 5 - 2 = 3 \] The largest possible value for the expression \( a - b \) is 3.
Since 3 is less than 12, there are no pairs \( (a, b) \) in \( A \times A \) for which \( a - b = 12 \).
Therefore, the relation \( R \) contains no elements. It is an empty set: \[ R = \emptyset \] Step 4: Final Answer:
The set \( R \) is the empty set, \( \emptyset \). So, option (A) is correct.