Question

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. R is the ”greater than or equal to” relation on the set of real numbers: For all x, y ∈ R , xRy ⇐⇒ x ≥ y 

Answer 1

Since for any $x \in R\,\,x \ge x$ , then $\forall x\,\,xRx$ and the relation is reflexive.

Let the conditions $xRy$ and $yRx$ then $x \ge y$ and $y \ge x$ . This is only possible if $x=y$ . But then the condition $\forall x,y\,\,\,xRy \Rightarrow yRx$ is not met. So, relation isn't symmetric.

Let $xRy$ and $yRz$ . Then $x \ge y$ and $y \ge z$ . But then $x \ge z$ . So, $\forall x,y,z\,\,xRy \wedge yRz \Rightarrow xRz$ and relation is transitive.

Answer: relation is reflexive, relation isn't symmetric, relation is transitive.