mathematics-class-11-unit-02-chapter-03-relations-and-functions-l-3-5-bg-vvriqi2e

### <Success>Example</Success>
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<ul>
<li>
Let A={Δ’s |Δ is a triangle in R2}
</li>
<li>
R = {(Δ1, Δ2) $\in$ A$\times$A |Δ1 is congruent to Δ2}
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</ul>
Show that, R is an equivalence relation.
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### <Success>Solution</Success>
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<ul>
<li>i) R is symmetric Δ1 is congruent to Δ2 <$\Rightarrow$ Δ2 is congruent to Δ1</li>
</ul>
ii) Every triangle is congruent to itself R is reflexive.
iii) Suppose (Δ1, Δ2) $\in$ R and (Δ2, Δ3)$\in$ R
<ul>
<li>
(Δ1, Δ2) $\in$ R $\Rightarrow$ Δ1 is congruent to Δ2 $\longrightarrow$ (i)
</li>
<li>
Similarly (Δ2, Δ3) $\in$ R $\Rightarrow$ Δ2 is congruent to Δ3 $\longrightarrow$ (ii)
</li>
<li>
(i) & (ii) $\Rightarrow$ Δ1 is congruent to Δ3 $\Rightarrow$ (Δ1, Δ3) $\in$ R
</li>
</ul>
$\therefore$ R is an equivalence relation.
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### <Success>Example</Success>
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Let A = $\mathbb{R}^2$ \ {(0, 0)}-punctured plane
R = {((x1, y1), (x2, y2)) $\in$ A$\times$A |there exists a non-zero scaler λ$\in$R such that (x1, y1) = λ (x2, y2)}
Show that R is an equivalence relation.
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### <Success>Solution</Success>
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i) Let ((x1, y1), (x2, y2))$\in$ R
$\Rightarrow$ there exists 0 ≠ λ $\in$ R such that (x1, y1) = λ(x2, y2)
$\Rightarrow$ $\frac{1}{\lambda}$ (x1, y1) = (x2, y2)
$\Rightarrow$ (x2, y2) = $\frac{1}{\lambda}$ (x1, y1)
$\Rightarrow$ ((x2, y2), (x1, y1))$\in$ R
$\therefore$ R is symmetric.
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