Step-by-step Solution:

The relation is given by \( a \sim b \) if \( a - 2b \equiv 0 \pmod 3 \). This simplifies to \( a \equiv 2b \pmod 3 \), which means \( a + b \equiv 3b \equiv 0 \pmod 3 \). Thus, the relation simplifies to: \( a + b \) is a multiple of 3. Let's check the properties: 1. Reflexive: \( a + a = 2a \). This is not always a multiple of 3 (e.g., if \( a = 1 \), \( 1+1=2 \) is not divisible by 3). So, it is not reflexive. 2. Symmetric: If \( a + b \) is divisible by 3, then \( b + a \) is also divisible by 3. So, it is symmetric. 3. Transitive: Suppose \( a \sim b \) and \( b \sim c \). Then \( a+b \) is a multiple of 3 and \( b+c \) is a multiple of 3. For example, \( 1 \sim 2 \) and \( 2 \sim 4 \). However, \( 1 + 4 = 5 \), which is not divisible by 3, so \( 1 \nsim 4 \). Thus, it is not transitive. Therefore, the relation is symmetric but neither transitive nor reflexive.