	Let R be a relation defined on N as aRb is 2a + 3b is a multiple of 5, a, b ∈N. Then R is
A. Not Reflexive		
B. Transitive but not Symmetric
C. Symmetric but not Transitive		
D. an Equivalence Relation


Let's analyze the given relation \( R \) on \( \mathbb{N} \) (the set of natural numbers) where \( aRb \) is defined as \( 2a + 3b \) being a multiple of 5. We need to determine which properties the relation \( R \) satisfies.

### 1. Reflexivity:
A relation \( R \) is reflexive if \( aRa \) holds for every \( a \in \mathbb{N} \). For reflexivity:
\[
2a + 3a = 5a
\]
Since \( 5a \) is clearly a multiple of 5 for all \( a \), the relation is reflexive.

### 2. Symmetry:
A relation \( R \) is symmetric if \( aRb \) implies \( bRa \). Let's check:
\[
aRb \implies 2a + 3b \text{ is a multiple of 5}.
\]
For symmetry, we need:
\[
bRa \implies 2b + 3a \text{ is a multiple of 5}.
\]
However, \( 2a + 3b \) being a multiple of 5 does not necessarily imply that \( 2b + 3a \) will also be a multiple of 5. Thus, the relation \( R \) is **not symmetric**.

### 3. Transitivity:
A relation \( R \) is transitive if \( aRb \) and \( bRc \) imply \( aRc \). Assume:
\[
2a + 3b \equiv 0 \pmod{5} \quad \text{and} \quad 2b + 3c \equiv 0 \pmod{5}.
\]
We need to check if this implies:
\[
2a + 3c \equiv 0 \pmod{5}.
\]
Adding the two congruences:
\[
2a + 3b + 2b + 3c = 2a + 5b + 3c \equiv 2a + 3c \equiv 0 \pmod{5}.
\]
Since \( 5b \) is always a multiple of 5, \( 2a + 3c \equiv 0 \pmod{5} \), meaning \( aRc \). Therefore, \( R \) is transitive.

### Conclusion:
The relation \( R \) is reflexive and transitive but not symmetric. Therefore, the correct answer is:

**B. Transitive but not Symmetric**

---

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the definition of an equivalence relation?
2. How do you prove whether a relation is reflexive?
3. Can a relation be symmetric and transitive but not reflexive?
4. What are the implications if a relation is only transitive?
5. How do counterexamples disprove symmetry in a relation?
6. What types of problems often involve checking symmetry and transitivity?
7. How does the definition of natural numbers (\(\mathbb{N}\)) impact relation properties?
8. What is the difference between partial orders and equivalence relations?

**Tip:** When analyzing relations, always start by checking reflexivity, then symmetry, and finally transitivity to systematically determine the nature of the relation.

Explore the relation R defined on natural numbers where aRb means 2a + 3b is a multiple of 5. Understand the concepts of reflexivity, symmetry, and transitivity in this context. Suitable for high school students and those interested in discrete mathematics.

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