What is the partition for a dice, where the universe is $\Omega=\{1, 2, 3, 4, 5, 6\}$?

Question

We launch a dice once. The dice has 6 possible results.

The universe is $\Omega=\{1, 2, 3, 4, 5, 6\}$.

What is the partition of the universe ?

Is it $\{1, 2, 3, 4, 5, 6\}$ ?

Is it $\{1\}$, $\{2\}$, ... $\{6\}$ ?

Is it $\{1\}$, $\{2\}$, ... $\{6\}$, $\{1, 2\}$, etc. $\{1, 2, 6\}$ ?

(Sorry for my basic question)

score 1 · Accepted Answer · answered Mar 14 '21 at 11:41

"A partition is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset." (Wikipedia)

There are many partitions of the same set. You have listed some of the possible ones:

$$\mathcal{P}_1 = \Big\{ \{1,2,3,4,5,6\}\Big\} $$ $$\mathcal{P}_2 = \Big\{ \{1\},\{2\},\{3\},\{4\},\{5\},\{6\} \Big\} $$ $$\mathcal{P}_3 = \Big\{ \{1,2\},\{3,4\},\{5,6\}\Big\} $$

But there are many others: $$\mathcal{P}_4 = \Big\{ \{1,4\},\{2\},\{3,5,6\}\Big\} $$ $$\mathcal{P}_5 = \Big\{ \{1\},\{2\},\{3,4,5,6\}\Big\} $$ $$\mathcal{P}_6 = \Big\{ \{1,2\},\{3\},\{4\},\{5,6\}\Big\} $$ and so on.

Thanks a lot for your huge help. I thought that a partition was made of only sets with only one element : this was my big problem. — Mathieu Krisztian, Mar 14 '21 at 11:42

What is the partition for a dice, where the universe is $\Omega=\{1, 2, 3, 4, 5, 6\}$?

1 Answers1