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What is a null or empty set?
What is a null set example?
How do you define a null set?
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 naรฏve settings). So the interval 
has size 
and the set 
has size 
.
In this setting, examples of non-empty null-sets are:
- Single-element sets, like
. - In fact, any countable set, like
- 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 
. 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 "
if and only if 
is a null set", then the null sets are exactly those equivalent to the empty set.
In measure theory, a null set refers to a set of measure zero. For example, in the reals, $\mathbb R$ with its standard measure (Lebesgue measure), the set of rationals $\mathbb Q$ has measure $0$, so $\mathbb Q$ is a null set in $\mathbb R$. Actually, all finite and countably infinite subsets of $\mathbb R$ have measure $0$. In contrast, the empty set always refers to the unique set with no elements, denoted $\left\{ \right\}$, $\varnothing$ or $\emptyset$.
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.
The only element of $\{\{\varnothing\}\}$ is $\{\varnothing\}$ which is in $\{\varnothing, \{\varnothing\}\}$.
Thus it is a subset...
a general rule: $$ a \in S \Rightarrow \{a\} \subset S $$ since, in your example $$ \{\emptyset\} \in \{\emptyset,\{\emptyset\}\} $$ the result follows