Determine and prove if the following are equivalence relations, partial ordering relations, or neither.

Question

{$(a, b) | a∈Z^{+}, b∈Z^{+}, a = 3^{n}b, where n∈N$} (N is the set of natural numbers)

{$(a, b) | a∈Z^{+}, b∈Z^{+}$, a >2b or b >2a}

{$(a, b) | a∈Z^{+}, b∈Z^{+}$, a ≡ 0 (mod b) or b ≡ 0 (mod a)}

Answer 1

All three relations are of the form $$aRb\quad \text{ if }\frac{a}{b}\in S$$ where $S$ is some subset of $[0,+\infty]$. Namely,

1. $S=\{3^n: n\ge 0\}$
2. $S=[0,1/2)\cup (2,\infty]$
3. $S=\mathbb N\cup \{1/x:x\in \mathbb N\}$

(One can observe that formally setting $0/0=1$ fits in all three cases.)

• Reflexivity amounts to $1\in S$.
• Symmetry amounts to $S=\{1/x:s\in S\}$.
• Antisymmetry (weak) amounts to $S\cap \{1/x:s\in S\}=\{1\}$
• Transitivity amounts to $S$ being closed under multiplication.