Question

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?

• A. Both \( \sigma_1 \) and \( \sigma_2 \) are equivalence relations
• B. Neither \( \sigma_1 \) nor \( \sigma_2 \) is an equivalence relation
• C. \( \sigma_1 \) is an equivalence relation, but \( \sigma_2 \) is not an equivalence relation
• D. \( \sigma_1 \) is not an equivalence relation, but \( \sigma_2 \) is an equivalence relation

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 \( \mathbb{R} \).

Answer 1

The Correct Option is A

Concept:
A relation is an equivalence relation if it satisfies three properties:
• Reflexive: \( a R a \) for all \( a \).
• Symmetric: If \( a R b \), then \( b R a \).
• Transitive: If \( a R b \) and \( b R c \), then \( a R c \).

Step 1: Analyze \( \sigma_1 \) (\( a-b \in \mathbb{Q} \)).

• Reflexive: \( a-a = 0 \). Since 0 is rational, \( a \sigma_1 a \).
• Symmetric: If \( a-b = q \in \mathbb{Q} \), then \( b-a = -q \). Since the negative of a rational is rational, \( b \sigma_1 a \).
• Transitive: If \( a-b = q_1 \) and \( b-c = q_2 \) (where \( q_1, q_2 \in \mathbb{Q} \)), then \( (a-b) + (b-c) = q_1 + q_2 \Rightarrow a-c = q_1 + q_2 \). Since the sum of rationals is rational, \( a \sigma_1 c \). So, \( \sigma_1 \) is an equivalence relation.

Step 2: Analyze \( \sigma_2 \) (\( a-b \in \mathbb{Z} \)).

• Reflexive: \( a-a = 0 \in \mathbb{Z} \).
• Symmetric: If \( a-b = k \in \mathbb{Z} \), then \( b-a = -k \in \mathbb{Z} \).
• Transitive: If \( a-b = k_1 \) and \( b-c = k_2 \) (where \( k_1, k_2 \in \mathbb{Z} \)), then \( a-c = k_1 + k_2 \). Since the sum of integers is an integer, \( a \sigma_2 c \). So, \( \sigma_2 \) is also an equivalence relation.

Step 3: Conclusion.
Both relations satisfy all three necessary properties. They are essentially congruences under the subgroups \( (\mathbb{Q}, +) \) and \( (\mathbb{Z}, +) \) within the group \( (\mathbb{R}, +) \).