Set theory: Basic Partitions question

Question

The example I'm using is from Wikipedia

The set { 1, 2, 3 } has these five partitions (one partition per item):

{ {1}, {2}, {3} }, sometimes written 1|2|3.

{ {1, 2}, {3} }, or 12|3.

{ {1, 3}, {2} }, or 13|2.

{ {1}, {2, 3} }, or 1|23.

{ {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).

---

Question:

If I take any of the 2-1 partitions 12|3, 13|2, or 1|23 and find the intersection with 1|2|3 isn't it non-empty? Therefore why is it a partition?

Answer 1

$\{ \{1\}, \{2\},\{3\}\}$ is a partition because $\{1\} \cup \{2\} \cup \{3\} = \{1,2,3\} $ and $\{1\} \cap \{2\}= \emptyset, \{1\} \cap \{3\}= \emptyset,\{2\} \cap \{3\}= \emptyset$.

Notice that the verification process doesn't involve another partition.

It is alright for two different partitions to intersect.

Answer 2

Isn't the third line referring to exactly that? I have two distinct elements of the partition (12|3 and 1|2|3 for example), and they are not equal, thus their intersection should be the null set? But their intersection is {3}, isn't it?

It is refering to the intersections of sets in a family, not the intersection of families.

Equivalently, a family of sets $P$ is a partition of $X$ if and only if all of the following conditions hold:

• The family $P$ does not contain the empty set (that is ${\displaystyle \emptyset \notin P}$ ).
• The union of the sets in $P$ is equal to $X$ (that is ${\displaystyle \textstyle \bigcup _{A\in P}A=X}$ ). The sets in $P$ are said to cover $X$.
• The intersection of any two distinct sets in $P$ is empty (that is ${\displaystyle (\forall A,B\in P)\;A\neq B\implies A\cap B=\emptyset }$). The elements of $P$ are said to be pairwise disjoint.

Here, $X = \{1,2,3\}, P_1=[{12\vert 3}], P_2=[{1\vert 2\vert 3}]$

The sets in the family of $[{12\vert 3}]$ are $\{1,2\},\{3\}$ and their intersection is empty, their union equals $X$, and the empty set is not included in the family. The family of $P_1$ is therefore a partition of $X$.

The sets in the family of $[{1\vert 2\vert 3}]$ are $\{1\},\{2\},\{3\}$ and the intersection of any pair of sets in that family is empty, their union equals $X$, and the empty set is not included in the family. The family of $P_2$ is therefore also a partition of $X$.

The families $P_1,P_2$ are both partitions of $X$. However, they are not themselves parts of a partition.