The probability that a relation R from <math> <mrow> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>}</mo> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>t</mi> <mi>o</mi> <mtext> </mtext> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>}</mo> </mrow> </mrow> </math> &nbsp;is both symmetric and transitive, is equal to

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Maths NCERT Exemplar Solutions Class 11th Chapter Three Maths Class 11th Relations and Functions

The probability that a relation R from ${x, y} t o {x, y}$ is both symmetric and transitive, is equal to

Option 1 -

$\frac{5}{16}$

Option 2 -

$\frac{9}{16}$

Option 3 -

$\frac{11}{16}$

Option 4 -

$\frac{13}{16}$

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

2 weeks ago

Correct Option - 1

Detailed Solution:

Total number of possible relation = $2^{n^{2}} = 2^{4} = 16$ $^{^{}}^{}$

Favourable relations = $?, {(x, x)}, {(y, y)}$

${(x, x), (y, y)}$

${(x, x), (y, y), (x, y), (y, x)}$

Probability = $\frac{5}{16}$

Total number of possible relation = <math> <mrow> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mn>6</mn> </mrow> </math> <math><mrow><msup><mrow></mrow><mrow><msup><mrow></mrow><mrow></mrow></msup></mrow></msup><msup><mrow></mrow><mrow></mrow></msup></mrow></math>Favourable relations = <math> <mrow> <mi>? </mi> <mo>, </mo> <mrow> <mo> {</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mo>, </mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>, </mo> <mrow> <mo> {</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mi>y</mi> <mo>, </mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </math> <math><mrow><mrow><mrow><mrow><mrow></mrow></mrow></mrow></mrow><mrow><mrow><mrow><mrow></mrow></mrow></mrow></mrow></mrow></math> <math> <mrow> <mrow> <mo> {</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mo>, </mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>, </mo> <mrow> <mo> (</mo> <mrow> <mi>y</mi> <mo>, </mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </math> <math> <mrow> <mrow> <mo> {</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mo>, </mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>, </mo> <mrow> <mo> (</mo> <mrow> <mi>y</mi> <mo>, </mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>, </mo> <mrow> <mo> (</mo> <mrow> <mi>x</mi> <mo>, </mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>, </mo> <mrow> <mo> (</mo> <mrow> <mi>y</mi> <mo>, </mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </math> Probability = <math> <mrow> <mfrac> <mrow> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>6</mn> </mrow> </mfrac> </mrow> </math>

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