Question

A relation R is defined in the set N as follows:
R = (x, y) : x = y - 3, y > 3
Then, which of the following is correct?

• A. (2, 4) \( \in \) R
• B. (5, 8) \( \in \) R
• C. (3, 7) \( \in \) R
• D. (1, 5) \( \in \) R

Hint:

For questions involving relations, carefully check all conditions for each option. It's easy to make a mistake by only checking one of the conditions. A systematic check, as shown above, prevents errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
An ordered pair (x, y) belongs to the relation R if and only if it satisfies both conditions given in the definition of R:
1. \( y>3 \)
2. \( x = y - 3 \)
We need to test each of the given options against these two conditions.
Step 2: Detailed Explanation or Calculation:
(A) (2, 4): Here, \( x = 2 \) and \( y = 4 \).
- Check condition 1: \( y>3 \). \( 4>3 \) is true.
- Check condition 2: \( x = y - 3 \). \( 2 = 4 - 3 \), which means \( 2 = 1 \). This is false.
So, (2, 4) \( \notin \) R.
(B) (5, 8): Here, \( x = 5 \) and \( y = 8 \).
- Check condition 1: \( y>3 \). \( 8>3 \) is true.
- Check condition 2: \( x = y - 3 \). \( 5 = 8 - 3 \), which means \( 5 = 5 \). This is true.
Since both conditions are satisfied, (5, 8) \( \in \) R.
(C) (3, 7): Here, \( x = 3 \) and \( y = 7 \).
- Check condition 1: \( y>3 \). \( 7>3 \) is true.
- Check condition 2: \( x = y - 3 \). \( 3 = 7 - 3 \), which means \( 3 = 4 \). This is false.
So, (3, 7) \( \notin \) R.
(D) (1, 5): Here, \( x = 1 \) and \( y = 5 \).
- Check condition 1: \( y>3 \). \( 5>3 \) is true.
- Check condition 2: \( x = y - 3 \). \( 1 = 5 - 3 \), which means \( 1 = 2 \). This is false.
So, (1, 5) \( \notin \) R.
Step 3: Final Answer:
The only pair that satisfies both conditions is (5, 8).