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Why do the subsets in a partition have to be nonempty?

Ragnarok

Member
Jul 10, 2013
41
"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!
 

Andrei

Member
Jan 18, 2013
36
The empty set does not belong to any set of equivalence classes.
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
"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 think it's just a matter of definition.

Certainly, if you don't follow this, many results about partitions will need to be tweaked.
There is nothing mathematically wrong in including the empty set also when considering partitions but that is not how it has been done.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Partitions are used to generate equivalence relations on a set, and vice versa.

It is sometimes useful to consider partitions of the empty set itself; intuitively one feels that this partition should be unique, which is only possible if there are NO possible empty sets in a partition (convince yourself that this indeed comprises the only partition possible in the definition you gave for an empty set X).
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
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.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
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.
Wrong kind of partition: the kind the OP is talking about might be a partition into singleton subsets, which would be fine....but such a partition would be rather unsuitable for integration.