In measure theory, a null set refers to a set of measure zero. For example, in the reals, 
with its standard measure (Lebesgue measure), the set of rationals 
has measure 
, so 
is a null set in 
. Actually, all finite and countably infinite subsets of 
have measure 
. In contrast, the empty set always refers to the unique set with no elements, denoted 
, 
or 
.
They aren't the same although they were used interchangeable way back when.
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set (or simply a measure-zero set). More generally, whenever an ideal is taken as understood, then a null set is any element of that ideal.
Whereas an empty set is defined as:
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set.
Why do we consider the null set to be a subset of all sets? Could we define it otherwise?
State: [] or null
What is a null or empty set?
What is an empty set example?
How do you write an empty set?
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From the Fundamental methods of mathematical economics (4th ed.) by Chiang and Wainwright, page 10:
“The smallest possible subset of S is a set that contains no elements at all. Such a set is called the null set, or empty set, denoted by the symbol Ø or {}.”
“The reasoning for considering the null set as a subset of S is quite interesting: If the null set is not a subset of S (Ø ⊄ S), then Ø must contain at least one element 𝑥 such that 𝑥 ∉ S. But since by definition the null set has no element whatsoever, we cannot say that Ø ⊄ S; hence the null set is a subset of S”
Question:
Why do we define a subset this way, leading to the inclusion of the null set? Could we not (more intuitively) define a subset of S: containing at least one element 𝑥 such that 𝑥 ∈ S AND no one element 𝑥 such that 𝑥 ∉ S?
My intuitive thinking:
If I have an apple, an orange, and a kiwi, I usually don’t also go around thinking that I also have a ‘no fruit’. Feels wrong to claim that ‘no element’ is a good description of my set that definitely contains elements.
Edit: Wow, THANK YOU everyone for such a robust discussion. Lots to think on, lots to turn over in my mind.
In analysis and measure theory, the term null set is also used to denote a set which has "size" zero, but in that case, size means a different thing. For instance, on the real line, it is customary to use length (at least in naïve settings). So the interval $[-3, 5]$ has size $8$ and the set $(2, 3)\cup [5, 7]$ has size $3$.
In this setting, examples of non-empty null-sets are:
- Single-element sets, like $\{5\}$.
- In fact, any countable set, like $\Bbb Q$
- The Cantor set
So whether there is a difference between the phrases "empty set" and "null set" depends entirely on the context.
This depends on the context.
In the context of set theory, the null set is the empty set. And that's the end of it.
In the context of measure theory, analysis, or probability, a null set is a set whose measure is $0$. For example in the usual Borel measure, finite sets are null sets; countable sets are null sets; and even some uncountable sets (e.g. the Cantor set) are null sets. But they are certainly not empty.
In that context, a null set is a set which is completely uninteresting "for practical purposes" and we can ignore safely ignore it if we choose to. So this statement is more general than just "empty".
Note, however, that if you define an equivalence relation "$A\sim B$ if and only if $A\mathbin{\triangle}B$ is a null set", then the null sets are exactly those equivalent to the empty set.