Question

On the set $R$ of real numbers we defined $xPy$ if and only if $xy\ge 0$. Then the relation $P$ is

• A. Reflexive but not symmetric
• B. Symmetric but not reflexive
• C. Transitive but not reflexive
• D. Reflexive and symmetric but not transitive

Answer 1

The correct option is D Reflexive and symmetric but not transitive

Let $P$ be the relation on the set of real numbers $R$ such that $xPy$ if and only if $xy\ge 0$

(i)

We know that, for any real number $x,x^2\ge 0$

$⟹xx\ge 0⟹xPx$

$\therefore$ $P$ is reflexive

(ii)

Let $(x,y)\in P$ i.e . $xPy$

$⟹xy\ge 0⟹yx\ge 0⟹yPx$

$\therefore P$ is symmetric

(iii)

Let $xPy$ and $yPz$

$⟹xy\ge 0$ and $yz\ge 0$

But from this, we can't conclude $xz\ge 0$

For example,

$(-1,0),(0,2)$ satisfies the relation $xy\ge 0$ but $(-1,2)$ doesn't satisfy relation $xy\ge 0$.

Thus, $P$ is not transitive.

Hence, $P$ is reflexive, symmetric but not transitive.