Question

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

• 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
$S=\left\{(x,y)\mid \frac{y}{2}\le x\le 2y\right\}$
Let $(x,x)\in S$, then
$\frac{x}{2}\le x\le 2x$
$\Rightarrow xSx$
$S$ is reflexive.
Now, let $(x,y)\in S$ such that $xSy$
$\Rightarrow \frac{y}{2}\le x\le 2y$
$\because \frac{y}{2}\le x$ and $x\le 2y$
$\Rightarrow y\le 2x$ and $\frac{x}{2}\le y$
$\Rightarrow \frac{x}{2}\le y\le 2x$
$\Rightarrow ySx$
$S$ is symmetric.
Note that
$(3,4)\in S\&(4,7)\in S$
but $(3,7)\notin S$
$S$ is not transitive.