33. Let R be a relation from N to N defined by R = {(a, b) : a, b $\in$ N and a = b²}. Are the following true?

(i) (a,a) $\in$ R, for all a _ N

(ii) (a,b) $\in$ R, implies (b,a) $\in$ R

(iii) (a,b) $\in$ R, (b,c) $\in$ R implies (a,c) $\in$ R.

Justify your answer in each case.

<p><strong>33. </strong>Given, R= { (a, b): a, b&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span> N and a = <em>b</em>²}</p><p><strong> (i) </strong>Let a = 2 <span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span>&nbsp;N</p><p>Then b = 2² = 4 <span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span>&nbsp;N</p><p>but a ≠ b.</p><p>Hence the given statement is not true.</p><p><strong> (ii)</strong> For a=<em>b</em>² the inverse b=<em>a</em>² may not hold true</p><p>Example (4,2)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span> R, a=4, b=2 and a=<em>b</em>²</p><p>but (2,4)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∉</mo></mrow></math></span>&nbsp;R.</p><p>Hence, the given statement is not true.</p><p><strong> (iii)</strong> If (a, b) <span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span>&nbsp;R</p><p>a=<em>b</em>²…… (1)</p><p>and (b, c)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span>&nbsp;R</p><p>b=<em>c</em>²……. (2)</p><p>so for (1) and (2), </p><p>a= (<em>c</em>²)²=<em>c</em>⁴.</p><p>is, a ≠<em>c</em>², </p><p>Hence, (a, c)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∉</mo></mrow></math></span>&nbsp;R.</p><p>? The given statement is false.</p>

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