Let R be a relation in a set N of natural numbers defined by R = {(a,b) = a is a mulatiple of b}. Then:

Options

A. `(2,5) in R`
B. `(-2,4)inR`
C. `(3,6)in R`
D. `(10,5)in R`

Answer

To determine which pairs belong to the relation R defined on the set of natural numbers N, where R = {(a, b) | a is a multiple of b}, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Relation**:
The relation R states that for a pair (a, b) to belong to R, a must be a multiple of b. This means there exists some integer k such that a = k * b.

2. **Evaluating Option 1: (2, 5)**:
- Here, a = 2 and b = 5.
- We need to check if 2 is a multiple of 5.
- Since there is no integer k such that 2 = k * 5, we conclude that 2 is not a multiple of 5.
- Therefore, (2, 5) does not belong to R.

3. **Evaluating Option 2: (-2, 4)**:
- Here, a = -2 and b = 4.
- However, -2 is not a natural number (the set of natural numbers N includes only positive integers).
- Since -2 is not in the set of natural numbers, (−2, 4) cannot belong to R.

4. **Evaluating Option 3: (3, 6)**:
- Here, a = 3 and b = 6.
- We need to check if 3 is a multiple of 6.
- Since there is no integer k such that 3 = k * 6, we conclude that 3 is not a multiple of 6.
- Therefore, (3, 6) does not belong to R.

5. **Conclusion**:
After evaluating all options, we find that none of the pairs belong to the relation R.