Let a relation R on N N be defined as: (x_1,y_1)R(x_2,y_2) if and only if x_1 x_2 or y_1 y…

Question

Let a relation $R$ on $N	imes N$ be defined as: $({x}_{1},{y}_{1})R({x}_{2},{y}_{2})$ if and only if ${x}_{1}\le {x}_{2}$ or ${y}_{1}\le {y}_{2}$.
Consider the two statements:

(I) $R$ is reflexive but not symmetric.

(II) $R$ is transitive

Then which one of the following is true?

Accepted Answer

$$\begin{aligned}
& \left(x_1, y_1\right) R\left(x_2, y_2\right) \
& \text { If } x_1 \leq x_2 \text { or } y_1 \leq y_2
\end{aligned}$$

For reflexive;

$$\begin{aligned}
& \left(x_1, y_1\right) R\left(x_1, y_1\right) \
& \Rightarrow x_1 \leq x_1 \text { or } y_1 \leq y_1
\end{aligned}$$

So, $R$ is reflexive

For symmetric

When $$\left(x_1, y_1\right) R\left(x_2, y_2\right)$$

$$\Rightarrow x_1 \leq x_2 \text { or } y_1 \leq y_2$$

For $$\left(x_2, y_2\right) R\left(x_1, y_1\right)$$

$$\Rightarrow x_2 \leq x_1 \text { or } y_2 \leq y_1$$

Not true for $(1,2)$ and $(3,4)$

For transitive

Take pairs as $(3,9),(4,6),(2,7)$

$(3,9) R(4,6)$

as $$4 \geq 3$$

$(4,6) R(2,7)$

As $$7 \geq 6$$

But $(3,9) R(2,7)$

As neither $$2 \geq 3$$ nor $$7 \geq 9$$

So not transitive

Solutions · 1