Question

Let R be the relation in the set N given by $R = \{(a, b) : a = b - 2, b>6\}$. Which of the following is the correct answer?

• A. $(2, 4) \in R$
• B. $(3, 8) \in R$
• C. $(6, 8) \in R$
• D. $(8, 7) \in R$

Hint:

Always test the easiest or most restrictive condition first. By simply looking for pairs where the second number $b$ is greater than 6, you immediately eliminate option (1) without any calculation. Then apply the equation test.

Answer 1

The Correct Option is C

To determine whether certain ordered pairs belong to the relation \( R = \{(a, b) : a = b - 2, b > 6\} \), we need to check each pair against the relation's criteria: \( a = b - 2 \) and \( b > 6 \).  

1. Check the pair \((2, 4)\):
   • The condition \( a = b - 2 \) translates to \( 2 = 4 - 2 \), which is true.
   • However, \( b = 4 \) does not satisfy \( b > 6 \) since \( 4 \not> 6 \).
   • Therefore, \((2, 4) \notin R\).
2. Check the pair \((3, 8)\):
   • The condition \( a = b - 2 \) translates to \( 3 = 8 - 2 \), which yields \( 3 = 6 \), not true.
   • \( b = 8 \) satisfies \( b > 6 \), but the first condition fails.
   • Therefore, \((3, 8) \notin R\).
3. Check the pair \((6, 8)\):
   • The condition \( a = b - 2 \) is \( 6 = 8 - 2 \), which simplifies to \( 6 = 6 \), true.
   • \( b = 8 \) does satisfy \( b > 6 \).
   • Both conditions are satisfied, hence \((6, 8) \in R\).
4. Check the pair \((8, 7)\):
   • The condition \( a = b - 2 \) is \( 8 = 7 - 2 \), which simplifies to \( 8 = 5 \), not true.
   • \( b = 7 \), however satisfies \( b > 6 \), but fails the first condition.
   • Thus, \((8, 7) \notin R\).

Based on the analysis, the only pair that satisfies both constraints \( a = b - 2 \) and \( b > 6 \) is \((6, 8)\). Therefore, the correct answer is \((6, 8) \in R\).