Question
Test whether the following relations R3 are:
- Reflexive.
- Symmetric.
- Transitive.
R3 on R
defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$R3 on R
defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$Then,
$\text{a}\in\text{R}_3$Implies that a2 - 4a2 + 3a2 = 0
So, R3 is reflexive.
Symmetry: Consider,
$\text{a, b}\in\text{R}_3$Implies that a2 - 4a2b2 + 3b2 = 0
But $\text{b}_2-4\text{b}_2\text{a}_2+3\text{a}_2\neq0$ for all $\text{a, b}\in\text{R}$
So, R3 is not symmetric.
Transitivity:
$1,2\in\text{R}_3$ and $2,3\in\text{R}_3$Implies that 1 - 8 + 6 = 0 and 4 - 24 + 27 = 0
But $1 - 12 + 9 \neq0$
So, R3 is not transitive.
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