Assignment -1
Grade Class 12
Question:
<p>Let S be the set of all real numbers and let R be a relation on S, defined by a Rb <span class="math-tex">\(\Leftrightarrow\)</span> (1 + ab) > 0. Then, R is</p>
<p style="display:inline">Reflexive and transitive but not symmetric</p>
<p style="display:inline">Symmetric and Reflexive but not transitive</p>
<p style="display:inline">reflexive and symmetric but not transitive</p>
<p style="display:inline">Symmetric and transitive but not reflexive</p>
Step-by-Step Solution
Key Concept: To determine the properties of a relation, verify reflexivity and symmetry through general algebraic identities while testing transitivity with specific numerical counterexamples, particularly involving negative values.
<p>Let S denote the set of all real numbers. Let R be a relation in S defined as a R b iff 1 + ab > 0.</p>
<ol start="1" style="list-style-type:lower-roman">
<li>R is reflexive, Let a be any real number.<br />
Then 1 + aa = 1 + a<sup>2</sup> > 0, since a<sup>2</sup> <span class="math-tex">\(\geq\)</span> 0.<br />
Thus a R a <s>v</s> a <span class="math-tex">\(\in\)</span> S. Therefore R is reflexive.</li>
<li>R is symmetric. Let a, b be any two real numbers.<br />
Then a R b <span class="math-tex">\(\Rightarrow\)</span> 1 + ab > 0 <span class="math-tex">\(\Rightarrow\)</span> 1 + ba > 0 [<span class="math-tex">\(\because\)</span> ab = ba]<br />
<span class="math-tex">\(\therefore\)</span> R is symmetric.</li>
<li>R is not transitive. Consider three real number 1, -<span class="math-tex">\(- \frac {1}{2}\)</span>, -4.<br />
We have <br />
<span class="math-tex">\(1+1\left(-\frac{1}{2}\right)=\frac{1}{2}>0\)</span><br />
<span class="math-tex">\(\therefore 1 R-\frac{1}{2}\)</span><br />
Further <span class="math-tex">\(1+\left(-\frac{1}{2}\right)(-4)=3>0\)</span><br />
<span class="math-tex">\(\therefore-\frac{1}{2} R-4\)</span><br />
But 1 + 1(-4) = -3 Which is not greater than 0. Therefore 1 is not R-related to -4.<br />
Thus 1 <span class="math-tex">\(R-\frac{1}{2},-\frac{1}{2} R-4\)</span> and 1 is not R-related to -4.<br />
<span class="math-tex">\(\therefore\)</span> R is not transitive.</li>
</ol>
Correct Answer: C