Relations
Equivalence Relations
AIEEE 2010
Grade 11
Question:
Consider: $R = \{(x, y) \mid x, y \in \mathbb{R}, x = wy \text{ for some rational } w\}$ and $S = \{(m/n, p/q) \mid m, n, p, q \in \mathbb{Z}, n, q \neq 0, qm = pn\}$. Then:
(1) $S$ is equivalence, $R$ is not
(2) $R$ and $S$ both equivalence
(3) $R$ equivalence, $S$ not
(4) Neither $R$ nor $S$ is equivalence
Step-by-Step Solution
Key Concept: $R$: $(0, 1) \in R$ (take $w = 0$) but $(1, 0) \notin R$ (no rational $w$ with $1 = 0 \cdot w$) $\Rightarrow$ not symmetric. $S$: $qm = pn \Leftrightarrow m/n = p/q$ — equality of rationals, which is an equivalence relation. Answer: (1).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)