Test whether the following relations R_1 are: 1. Reflexive. 2. Sy… - Vidyadip

Question

Test whether the following relations $R_1$ are:

Reflexive.
Symmetric.
Transitive.

$R_1$ on $Q_0$ defined by $(\text{a, b})\in\text{R}_1\Leftrightarrow\ \text{a}=\frac{1}{\text{b}}.$

✓

Answer

$\text{R}_1=\Big\{(\text{x, y}),\ \text{x, y}\in\text{Q}_0, \text{x}=\frac{1}{\text{y}}\Big\}$
Reflexivity: Let, $\text{x}\in\text{Q}_0$
$\Rightarrow\ \text{x}\neq\frac{1}{\text{x}}$
$\Rightarrow\ (\text{x, x})\in\text{R}_1$
$\therefore R_1$ is not reflexive.
Symmetric: Let, $(\text{x, y})\in\text{R}_1$
$\Rightarrow\ \text{x}=\frac{1}{\text{y}}$
$\Rightarrow\ \text{y}=\frac{1}{\text{x}}$
$\Rightarrow\ (\text{y, x})\in\text{R}_1$
$\therefore R_1$ is Symmetric.
Transitive: Let, $(\text{x, y})\in\text{R}_1$ and $(\text{y, z})\in\text{R}_1$
$\Rightarrow\ \text{x}=\frac{1}{\text{y}}$ and $\text{y}=\frac{1}{\text{z}}$
$\Rightarrow\ \text{x}=\text{z}$
$\Rightarrow\ (\text{x, z})\notin\text{R}_1$
$\therefore R_1$ is not Transitive.

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