First slide

Relations XII

Question

Let a relation R on the set N of natural number be defined as $(x, y) \in R$ If and only If $x^{2} - 4 xy + 3 y^{2} = 0$ for all, $x, y \in N$ and the relation is

Moderate

A

reflexive
B

symmetric
C

transitive
D

an equivalence relation

Solution

we have, $R = {(x, y); x^{2} - 4 xy + 3 y^{2} = 0, x, y \in N$
Let $x \in N, x^{2} - 4 x \cdot x + 3 x^{2} = 0$
$∴ (x, x) \in R$
R is reflexive
we have,
$\begin{aligned} (3)^{2} - 4 (3) (1) + 3 (1)^{2} = 9 - 12 + 3 = 0 \\ (3, 1) \in R \end{aligned}$
Also, $(1)^{2} - 4 (1) (3) + 3 (3)^{2} = 1 - 12 + 27 = 16 \neq 0$
$(3)^{2} - 4 (3) (1) + 3 (1)^{2} = 0$
now, $(9, 1) \in R if (9)^{2} - 4 (9) (1) + 3 (1)^{2} = 0$
i.e $48 \neq 0$
which is not so $(9, 3), (3, 1) \in R and (9, 1) \notin R$
$∴$ R is not transitive.

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