Test whether the following relations R_3 are: 1. Reflexive. 2. Sy… - Vidyadip

Question

Test whether the following relations $R_3$ are:

Reflexive.
Symmetric.
Transitive.

$R_3$ on $R$ defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$

✓

Answer

Reflexivity: Consider a be an arbitrary element of $R_3$ Then, $\text{a}\in\text{R}_3$
Implies that $a_2 - 4a_2 + 3a_2 = 0$
So, $R_3$ is reflexive.
Symmetry: Consider, $\text{a, b}\in\text{R}_3$
Implies that $a_2 - 4a_2b_2 + 3b_2 = 0$
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, $R_3$ 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, $R_3$ is not transitive.

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