Question
Test whether the following relations $R_2$ are:
- Reflexive.
- Symmetric.
- Transitive.
✓
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.
$\Rightarrow\ |\text{a}-\text{a}|=0\leq5$
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.
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