Question

Let $S$ be the relation on set of positive real number defined by $S=\left\{(x, y) \mid \frac{y}{2} \leq x \leq 2 y\right\}$

• A. Reflexive only
• B. Transitive only
• C. Symmetric only
• D. Reflexive and symmetric only

Answer 1

Answer: (D)

Here, the relation on set of positive real numbers is
$S=\left\{(x, y) \mid \frac{y}{2} \leq x \leq 2 y\right\}$
Let $(x, x) \in S$, then
$\frac{x}{2} \leq x \leq 2 x$
$\Rightarrow x S x$
$S$ is reflexive.
Now, let $(x, y) \in S$ such that $x S y$
$\Rightarrow \frac{y}{2} \leq x \leq 2 y$
$\because \frac{y}{2} \leq x$ and $x \leq 2 y$
$\Rightarrow y \leq 2 x$ and $\frac{ x }{2} \leq y$
$\Rightarrow \frac{x}{2} \leq y \leq 2 x$
$\Rightarrow y S x$
$S$ is symmetric.
Note that
$(3,4) \in S \&(4,7) \in S$
but $(3,7) \notin S$
$S$ is not transitive.