In a set of real numbers a relation R is defined as x R y such that $| x | + | y | \leq \frac{1}{2},$ then relation R is

Option 1 -

Reflexive and symmetric but not transitive

Option 2 -

Symmetric but not transitive and reflexive

Option 3 -

Transitive but not symmetric and reflexive

Option 4 -

None of reflexive, symmetric and transitive

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 2

Detailed Solution:

$| x | + | y | \leq \frac{1}{2}$

If x = y, $2 | x | \leq \frac{1}{2}, - \frac{1}{4} \leq x \leq \frac{1}{4}$

$∴ (x, x) \notin R v x \in r e a l n o$

$∴$ R is not reflexive

$I f | x | + | y | \leq \frac{1}{2} \Rightarrow | y | + | x | \leq \frac{1}{2}$

$∴ (x, y) \in R \Rightarrow (y, x) \in R$

$∴$ R is symmetric

$I f | x | + | y | \leq \frac{1}{2} a n d | y | + | z | \leq \frac{1}{2}$

$\Rightarrow | x | + | z | \leq \frac{1}{2}$

$∴$ R is not transitive.

<math> <mrow> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>y</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>             If x = y,   <math> <mrow> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math>   <math> <mrow> <mo>∴</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>∉</mo> <mi>R</mi> <mtext> </mtext> <mtext> </mtext> <mi>v</mi> <mtext> </mtext> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> <mtext> </mtext> <mo>∈</mo> <mtext> </mtext> <mtext> </mtext> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> <mtext> </mtext> <mtext> </mtext> <mi>n</mi> <mi>o</mi> </mrow> </math>              <math> <mrow> <mo>∴</mo> </mrow> </math>         R is not reflexive <math> <mrow> <mi>I</mi> <mi>f</mi> <mtext> </mtext> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>y</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>⇒</mo> <mrow> <mo>|</mo> <mrow> <mi>y</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>              <math> <mrow> <mo>∴</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mi>R</mi> <mtext> </mtext> <mo>⇒</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mi>R</mi> </mrow> </math>           <math> <mrow> <mo>∴</mo> </mrow> </math> R is symmetric  <math> <mrow> <mi>I</mi> <mi>f</mi> <mtext> </mtext> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>y</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mi>y</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <menclose notation="updiagonalstrike"> <mo>⇒</mo> </menclose> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>             <math> <mrow> <mo>∴</mo> </mrow> </math> R is not transitive.

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In a set of real numbers a relation R is defined as x R y such that $| x | + | y | \leq \frac{1}{2},$ then relation R is