Question about partitions with a single element and equivalence relations

Question

I couldn't find a formal definition of a partition but I found this picture on the Bell numbers wiki.

You can see there are no partitions with a single element, it confused me, why a partition with a single element seemingly isn't considered a partition?

For example in the case of 5 elements, why doesn't $\{x\}$ or $\{\{x,y\},\{z\}\}$ considered partitions ?

Answer 1

"You can see there are no partitions with a single element"?? I see that the last partition in your picture does have a single element, namely the set of all five elements.

In more detail: If the 5 elements of the set that you're partitioning are named 1,2,3,4,5, then the first partition in your picture is $\{\{1\},\{2\},\{3\},\{4\},\{5\}\}$, which has 5 elements, each of which is a one-element set. The last partition in your picture is $\{\{1,2,3,4,5\}\}$, which has just one element, and that one element is a five-element set.

Concerning the last sentence of your question, $\{\{x,y\},\{z\}\}$ is a partition, but it is a partition of a 3-element set, $\{x,y,z\}$, not of a 5-element set. Since a partition is always a set of sets, $\{x\}$ would be a partition only if $x$ is a set, and then $\{x\}$ is a partition of the set $x$.

Answer 2

I'm not sure how exactly the figure relates to partitions... A partition of a set is a collection of subsets such that any element of the set belongs to exactly one subset. There is nothing forbidding one (or all) of those subsets from having only one element.