Relations
Types of Relations
Premium Question
Grade 11
Question:
The relation $R$ on $\mathbb{Z}$ defined by $(a, b) \in R \Leftrightarrow |a - b|$ is a perfect square is:
(1) reflexive and symmetric but not transitive
(2) an equivalence relation
(3) symmetric only
(4) reflexive only
Step-by-Step Solution
Key Concept: Reflexive: $|a - a| = 0 = 0^2\checkmark$. Symmetric: $|a - b| = |b - a|\checkmark$. Not transitive: $|0 - 1| = 1 = 1^2$, $|1 - 2| = 1 = 1^2$, but $|0 - 2| = 2$ (not a perfect square). Answer: (1).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)