Question

Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive.

(x, y) ∈ R if x ≥ y, where R is the set of positive integers.

Answer 1

Let us determine whether the relation $(x, y) ∈ R$  if $x ≥ y$ is reflexive, symmetric, antisymmetric and/or transitive.

Since $x\ge x$ for each positive integer $x,$ we conclude that the relation is reflexive.

Taking into account that $(2,1)\in R$ but $(1,2)\notin R,$ we conclude that the relation is not symmetric.

If $(x,y)\in R$ and $(y,x)\in R,$ then $x\ge y$ and $y\ge x.$ It follows that $x\ge y\ge x,$ and hence  $y=x.$We conclude that the relation is antisymmetric.

Since $(x,y)\in R$ and $(y,z)\in R$ imply $x\ge y$ and $y\ge z,$ we conclude that $x\ge y\ge z$, and thus  $x\ge z.$ It follows that $(x,z)\in R,$ and thus the relation is transitive.