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Definition of Reflexive relati...

# Definition of Reflexive relation

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<h2><span style="font-size:13pt"><span style="font-family:Cambria,serif"><span style="color:#365f91"><strong><span style="font-size:14.0pt">What is the definition of Reflexive relation?</span></strong></span></span></span></h2> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Reflexive relation onset happens to be a binary element where every element is related to itself.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Let us assume A to be a set and R to be the relation defined in it.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">R happens to be reflexive, if (a, a) </span><span style="color:black">&isin;</span><span style="color:black"> R for all a </span><span style="color:black">&isin;</span><span style="color:black"> A that is, all the elements of A is R-related to itself. Thus it can be said that aRa for every a </span><span style="color:black">&isin;</span><span style="color:black"> A.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">A relation &lsquo;R&rsquo; in a set &lsquo;A&rsquo; is not reflexive if there is at least an element a </span><span style="color:black">&isin;</span><span style="color:black"> A, in a form that (a, a) </span><span style="color:black">&notin;</span><span style="color:black"> R.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Consider, a set &lsquo;A&rsquo; = {p, q, r, s}.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">The relation R11 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since each element in set A is R11 - related to itself.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">But the relation R22 = {(p, p), (p, r), (q, r), (q, s), (r, s)} is not reflexive in A, as q, r, s </span><span style="color:black">&isin;</span><span style="color:black"> A, but (q, q) </span><span style="color:black">&notin;</span><span style="color:black"> R22, (r, r) </span><span style="color:black">&notin;</span><span style="color:black"> R22 and (s, s) </span><span style="color:black">&notin;</span><span style="color:black"> R22</span></span></span></span></p> <h2><span style="font-size:13pt"><span style="font-family:Cambria,serif"><span style="color:#365f91"><strong><span style="font-size:14.0pt">Solved examples of Reflexive Relation on Set</span></strong></span></span></span></h2> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">1.</span></strong><span style="color:black"> A relation &lsquo;R&rsquo; is defined on the set Z (set of all integers) by &ldquo;aRb if and only if 2a + 3b is divisible by 5&rdquo;, for all a, b </span><span style="color:black">&isin;</span><span style="color:black"> Z. Find out whether R is a reflexive relation on Z.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">Solution:</span></strong></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Let a </span><span style="color:black">&isin;</span><span style="color:black"> Z. Now 2a + 3a = 5a, which is divisible by 5. So, it can be seen that aRa holds for all a in Z. In other words, R is reflexive.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">2.</span></strong><span style="color:black"> On the set Z, a relation R is defined by &ldquo;aRb. Find out if R is a reflexive relation on Z if a &ndash; b is divisible by 5&rdquo; for a, b </span><span style="color:black">&isin;</span><span style="color:black"> Z,</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">Solution:</span></strong></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Let a </span><span style="color:black">&isin;</span><span style="color:black"> Z. Here a &ndash; a is divisible by 5. So, here it can be found that aRa holds for all a in Z i.e. R is reflexive.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">3.</span></strong><span style="color:black"> Consider the set Z in which a relation R is defined by &lsquo;aRb if and only if a + 3b is divisible by 4, for a, b </span><span style="color:black">&isin;</span><span style="color:black"> Z. Prove that R is a reflexive relation on set Z.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">Solution:</span></strong></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">Let a </span><span style="color:black">&isin;</span><span style="color:black"> Z. Now a + 3a = 4a, which is divisible by 4. So, here, aRa holds for all a in Z i.e. R is reflexive.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">4.</span></strong><span style="color:black"> On the set of all real numbers R, a relation &rho; is defined by &lsquo;x&rho;y&rsquo; if and only if |x &ndash; y| &le; y, for x, y </span><span style="color:black">&isin;</span><span style="color:black"> R. Prove that the &rho; is not reflexive relation.</span></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><strong><span style="color:black">Solution:</span></strong></span></span></span></p> <p><span style="font-size:11pt"><span style="background-color:white"><span style="font-family:Calibri,sans-serif"><span style="color:black">The relation &rho; is not reflexive as x = -2 </span><span style="color:black">&isin;</span><span style="color:black"> R, but |x &ndash; x| = 0 which is not less than -2(= x). </span></span></span></span></p>

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Class 12th

Relations
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