Fine partitions

Question

I am tasked with the following:

Give four different partitions $\Pi_1,\Pi_2,\Pi_3,\Pi_4$ of the set $\Bbb N$ with $\Pi_i$ Finer that $\Pi_{i+1}$ for $i =1,2,3$

I think that partition by 8, 4,2 and 1 is an appropriate partition scheme. However I am unsure how to represent this scheme notation wise.

Answer 1

Well I think you're along the right path let's first identify each partition by $\Pi_a,\Pi_b,\Pi_c,\Pi_d$:

1. Partitioning by 8's: $$ \Pi_a = \{ \{ 8n + k \mid k = 0,1,\ldots,7 \} \mid n \in \mathbb{N} \} $$
2. Partitioning by 4's: $$ \Pi_b = \{ \{ 4n + k \mid k = 0,1,\ldots,3 \} \mid n \in \mathbb{N} \} $$
3. Partitioning by 2's: $$ \Pi_c = \{ \{ 2n + k \mid k = 0,1 \} \mid n \in \mathbb{N} \} $$
4. Partitioning by 1's: $$ \Pi_d = \{ \{ n \} \mid n \in \mathbb{N} \} $$

Now remember that a partition $\Pi$ is finer than another partition $\Omega$ if $$ \forall \pi \in \Pi \; \exists \; \omega \in \Omega \text{ s.t. } \pi \subset \omega $$ or in other words $\Pi$ is finer than $\Omega$ if every set in $\Pi$ is a subset of some set in $\Omega$. Thus we can actually see that $\Pi_d$ is the finest here. Do you see how to put $\Pi_1,\Pi_2,\Pi_3,\Pi_4$?

Answer 2

Alternative:

$\Pi_{1}=\left\{ A,B,C,D\right\} $, $\Pi_{2}=\left\{ A,B,C\cup D\right\} $, $\Pi_{3}=\left\{ A,B\cup C\cup D\right\} $, $\Pi_{4}=\left\{ A\cup B\cup C\cup D\right\} $

This for disjoint non-empty subsets $A$, $B$, $C$, $D$ of $\mathbb{N}$ that satisfy $A\cup B\cup C\cup D=\mathbb{N}$.