Relations
Equivalence Relations
AIEEE 2011
Grade 11
Question:
Statement I: $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : y - x \in \mathbb{Z}\}$ is an equivalence relation on $\mathbb{R}$.
Statement II: $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x = \alpha y \text{ for some rational } \alpha\}$ is an equivalence relation on $\mathbb{R}$.
(1) I true, II false
(2) I false, II true
(3) Both true
(4) Both false
Step-by-Step Solution
Key Concept: $A$: $y - x \in \mathbb{Z} \Rightarrow$ reflexive ($0 \in \mathbb{Z}$), symmetric ($-(y - x) \in \mathbb{Z}$), transitive (sum of integers). Equivalence $\checkmark$. $B$: $(0, 1) \in B$ ($\alpha = 0$) but $(1, 0) \notin B$ ($1 = \alpha \cdot 0$ has no solution). Not symmetric $\Rightarrow$ not equivalence. Answer: (1).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)