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 confirm if a relation qualifies as an equivalence relation, we must verify its reflexivity, symmetry, and transitivity.

Step 1: Assess reflexivity by examining if the sum of any integer with itself, \( x + x \), results in an even number for all integers \( x \). 

Step 2: Evaluate symmetry by confirming that if the sum of two integers, \( x + y \), is even, then the sum with their order reversed, \( y + x \), is also even. 

Step 3: Test transitivity by ensuring that if the sums \( x + y \) and \( y + z \) are both even, then the sum \( x + z \) is also even. 

Final Conclusion: The relation is confirmed as an equivalence relation, corresponding to Option 4.