Question
Test whether the following relations R2 are:
- Reflexive.
- Symmetric.
- Transitive.
R2 on Z
defined by $(\text{a, b})\in\text{R}_2\Leftrightarrow\ |\text{a}-\text{b}|\leq5$R2 on Z
defined by $(\text{a, b})\in\text{R}_2\Leftrightarrow\ |\text{a}-\text{b}|\leq5$$\text{a}\in\text{R}_2$
$\Rightarrow\ |\text{a}-\text{a}|=0\leq5$
So, R2 is reflexive.
Symmetry: Let
$(\text{a, b})\in\text{R}_2$$\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, R2 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, R2 is transitive.
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R3 on R
defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$