Let a set A = A1 $\cup$ A2 $\cup$ …. $\cup$ Ak, where Ai $\cap$ Aj = $?$ for $i\ne j,1\le i,j\le k.$ Define the reaction R from A to A by R = $\left\{\left(x,y\right):y\in A_{i}ifandonlyifx\in A_i,1\le i\le k\right\}.$ Then, R is:

<p>Use aRb = a is related to b, belongs to A iff a belongs to A.</p><p>In simple terms, aRb is true if both a &amp; b belongs to the same set.</p><p>For reflexive</p><p>aRa, a<span contenteditable="false"> <math> <mrow> <mo>∈</mo> </mrow> </math> </span> &lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1814274689"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;A, so it is true.</p><p>For symmetric</p><p>Let aRb be true</p><p>Þ a &amp; b belongs to the same set.</p><p>Þ b &amp; a also belongs to the same set</p><p>Þ bRa will be true</p><p>For transitive</p><p>Let aRb and bRc be true.</p><p>aRb Þ a, b belongs to the same set</p><p>bRc Þ b, c belongs to the same set</p><p>Þ (a, c) belongs to the same set</p><p>Þ so aRc will be true.</p><p>So R is an equivalence relation.</p>

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