Test whether the following relations R_2 are: 1. Reflexive. 2. Symmetric. 3. Transitive. R_2 on Z defined by (a, b) _2 |a-b| 5

Reflexivity: Let a be an arbitrary element of R_2. Then,a _2 |a-a|=0 5 So, R_2 is reflexive. Symmetry: Let (a, b) _2 |a-b| 5 |b-a| 5 [Since, |a - b| = |b - a|] (b, a) _2 So, R_2 is symmetric. Transitivity: Let (1, 3) _2 and (3,7) _2 |1-3| 5 and |3-7| 5 But |1-7| 5 (1,7) _2 So, R_2 is transitive.

Test whether the following relations R_2 are: 1. Reflexive. 2. Sy…

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Question

Test whether the following relations $R_2$ are:

Reflexive.
Symmetric.
Transitive.

$R_2$ on Z defined by $(\text{a, b})\in\text{R}_2\Leftrightarrow\ |\text{a}-\text{b}|\leq5$

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Answer

Reflexivity: Let a be an arbitrary element of $R_2$. Then,$\text{a}\in\text{R}_2$
$\Rightarrow\ |\text{a}-\text{a}|=0\leq5$
So, $R_2$ is reflexive.
Symmetry: Let $(\text{a, b})\in\text{R}_2$
$\Rightarrow\ |\text{a}-\text{b}|\leq5$
$\Rightarrow\ |\text{b}-\text{a}|\leq5$ [Since, |a - b| = |b - a|]
$\Rightarrow\ (\text{b, a})\in\text{R}_2$
So, $R_2$ is symmetric.
Transitivity: Let $(1, 3)\in\text{R}_2$ and $(3,7)\in\text{R}_2$
$\Rightarrow\ |1-3|\leq5$ and $|3-7|\leq5$
But $|1-7|\nleq5$
$\Rightarrow\ (1,7)\notin\text{R}_2$
So, $R_2$ is transitive.