Let a set A = A1 $\cup$ A2 $\cup$ ..... $\cup$ Ak, where Ai $\cap$ Aj = $\phi$ for i $\ne$ j, 1 $\le$ j, j $\le$ k. Define the relation R from A to A by R = {(x, y) : y $\in$ Ai if and only if x $\in$ Ai, 1 $\le$ i $\le$ k}. Then, R is :
Solution
<p>$R = \{ (x,y):y \in {A_i},\,iff\,x \in {A_i}\,1 \le i \ge k\}$</p>
<p>(1) Reflexive</p>
<p>(a, a) $\Rightarrow$ $a \in {A_i}$ iff $a \in {A_i}$</p>
<p>(2) Symmetric</p>
<p>(a, b) $\Rightarrow$ $a \in {A_i}$ iff $b \in {A_i}$</p>
<p>(b, a) $\in$R as $b \in {A_i}$ iff $a \in {A_i}$</p>
<p>(3) Transitive</p>
<p>(a, b) $\in$R & (b, c) $\in$R.</p>
<p>$\Rightarrow$ $a \in {A_i}$ iff $b \in {A_i}$ & $b \in {A_i}$ iff $c \in {A_i}$</p>
<p>$\Rightarrow$ $a \in {A_i}$ iff $c \in {A_i}$</p>
<p>$\Rightarrow$ (a, c) $\in$ R.</p>
<p>$\Rightarrow$ RElation is equivalnece.</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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