Question

The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is:

• A. reflexive and transitive but not symmetric
• B. reflexive and symmetric but not transitive
• C. symmetric and transitive but not reflexive
• D. an equivalence relation

Hint:

When dealing with relations, always verify the properties of reflexivity, symmetry, and transitivity to determine equivalence.

Answer 1

The Correct Option is D

To establish if the relation is an equivalence relation, we must verify reflexivity, symmetry, and transitivity.
Step 1: To confirm reflexivity, we check if \( x + x \) is even for every integer \( x \). 
Step 2: For symmetry, we confirm that if \( x + y \) is even, then \( y + x \) must also be even. 
Step 3: To verify transitivity, we ensure that if \( x + y \) and \( y + z \) are both even, then \( x + z \) is also even. 

Final Conclusion: The relation meets all criteria for an equivalence relation, corresponding to Option 4.