Define two relations \( \sigma_1 \) and \( \sigma_2 \) on the set of all real numbers \( \mathbb{R} \) as follows:
\( a \sigma_1 b \iff a - b \) is a rational number
\( a \sigma_2 b \iff a - b \) is an integer
Which one of the following is correct?
Show Hint
Any relation defined as \( aRb \iff a-b \in S \), where \( S \) is a subgroup of the reals under addition, will always be an equivalence relation. Both \( \mathbb{Q} \) and \( \mathbb{Z} \) are subgroups of \( \mathbb{R} \).