Relations & Functions
Equivalence Relations
Grade 12
Question:
<p>Consider the following relations. \(R = \{(x, y) | x, y \text{ are real numbers and } x = wy \text{ for some rational number } w\}\) and \(S = \left\{\frac{m}{p}; \frac{n}{q} \mid m, n, p, q \text{ are integers such that } n, q > 0 \text{ and } qm = pn\right\}\), then</p>
<p>(a) neither R nor S is an equivalence relation</p>
<p>(b) S is an equivalence relation but R is not an equivalence relation</p>
<p>(c) R and S both are equivalence relations</p>
<p>(d) R is an equivalence relation but S is not an equivalence relation</p>
Step-by-Step Solution
Key Concept: Both relations are equivalence relations; verify all three properties for each relation carefully
<p><strong>Analysis:</strong> For R: Reflexivity: $x = 1 \cdot x$ ✓. Symmetry: If $x = wy$, then $y = \frac{x}{w}$ where $\frac{1}{w}$ is rational ✓. Transitivity: If $x = w_1y$ and $y = w_2z$, then $x = w_1w_2z$ where $w_1w_2$ is rational ✓. For S: The relation $qm = pn$ represents equality of fractions. Reflexivity: $\frac{m}{p} = \frac{m}{p}$ ✓. Symmetry: If $\frac{m}{p} = \frac{n}{q}$, then $\frac{n}{q} = \frac{m}{p}$ ✓. Transitivity: If $\frac{m}{p} = \frac{n}{q}$ and $\frac{n}{q} = \frac{r}{s}$, then $\frac{m}{p} = \frac{r}{s}$ ✓.</p><p>∴ Answer is (c).</p>
Correct Answer: c