Question

Is the `divides’ relation on the set of positive integers
(a) reflexive? (b) symmetric? (c) transitive? (d) an equivalence relation?
Justify your answers.

Answer 1

(a) such relation is reflexive, because for any positive integer $x$, it is true that it is divisible by itself.

(b) such relation is not symmetric. For example, 2 divides 4, but 4 does not divide 2. Moreover, this relation is antisymmetric, because if $x$ divides $y$ and $y$ divides $x$ , then $x=y$.

(c) this relation is transitive. If $x$ divides $y$ , then $y=cx$ . If $y$ divides $z$, then $z=dy.$ Therefore, $z=dy=d(cx) = (dc)x$ , so $x$ divides $z$ .

(d) the relation is not an equivalence relation, because the equivalence relation needs to be symmetric (https://en.wikipedia.org/wiki/Equivalence_relation)