I think it's just a matter of definition.Ragnarok said:"A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets."
I was just wondering why the subsets must be nonempty. Is it just convention/convenient or is it because it would violate something else?
Thanks!
I like Serena said:Typically partitions are used to define a Riemann or Lebesgue integral.
For that to work, we need to be able to define a limit where the partition becomes infinitely fine grained, meaning each subset in the partition should be of size greater than or equal to $\varepsilon >0$.
In other words: not empty.
The subsets in a partition cannot be empty because a partition, by definition, is a collection of nonempty subsets that cover the entire set. If one or more subsets were empty, then they would not cover the entire set and the partition would not be valid.
The purpose of having nonempty subsets in a partition is to ensure that every element in the original set is accounted for in at least one of the subsets. This is important for accurately representing the entire set and its relationships with its subsets.
No, a partition cannot have subsets that are both empty and nonempty. As mentioned earlier, a partition must consist of nonempty subsets that cover the entire set. Having subsets that are both empty and nonempty would violate this requirement.
If a partition has only one nonempty subset, then it essentially becomes the same as the original set. This defeats the purpose of creating a partition in the first place, which is to break down the set into smaller, more manageable subsets.
No, there are no exceptions to this rule. The definition of a partition states that it must consist of nonempty subsets, and this rule must be followed for a partition to be considered valid.