# Difference between Null set and empty set

One of my friend asked this doubt.Even in lower class we use both as synonyms,he says that these two concepts have difference.Empty set $\{ \}$ is a set which does not contain any elements,while null set ,$\emptyset$ says about a set which does not contain any elements.

I could not make out that...is his argument correct ? if so how ?

• Well, sometimes "null set" may refer to a set of measure zero. Is this what you are after? In this case, a null set need not be an empty set but an empty set must be a null set. – Gary Moore Aug 13 '15 at 10:43

## 3 Answers

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.

I would call "null sets" in the Measure Theory sense "sets of measure 0" just to avoid any confusion. In many parts of mathematics, "null set" and "empty set" are synonyms.