Distinguishing equivalences and orders

Statement

Among the following relations on $\mathbb{N}$, distinguish the equivalences from the orders: (a) $a-b$ is even; (b) $a\le b$; (c) $a$ and $b$ have the same final digit; (d) $a$ divides $b$.

Solution

• (a) $a-b$ even → equivalence. It is reflexive ($a-a=0$ even), symmetric (if $a-b$ is even so is $b-a$) and transitive. It groups the numbers by parity.
• (b) $a\le b$ → order (total). Reflexive, antisymmetric, transitive, and every pair is comparable.
• (c) same final digit → equivalence. Reflexive, symmetric, transitive; the classes are ten (final digit $0,1,\ldots,9$).
• (d) $a$ divides $b$ → order (partial). Reflexive, antisymmetric and transitive, but not total: $2$ and $3$ are not comparable.

Links

Topics: Set theory
Concepts: Order relation · Equivalence relation
Skills: Reasoning by cases
Exercise type: Proof