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What is the Empty Set in Mathematics?
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.
Perhaps what you find confusing is the use of set-builder notation to define $P, Q, R$: Included in between { ... } are the condition(s) that any "candidate" element must satisfy in order to be included in the set, and a set defined by set-builder notation contains all, and only, those elements satisfying all the conditions given.
In each of $P,\; Q, \;R$, set-builder notation is used to provide the conditions for inclusion in each set, respectively. Note: unless otherwise stipulated, you can take conditions separated by a comma to be a conjunction of conditions; that is: $$X = \{x : \text{(condition 1), (condition 2), ...., (condition n)}\}$$ means $X$ is the set of all x such that x satisfies (condition 1) AND x satisfies (condition 2) AND ... AND x satisfies (condition n).
$$P = \{x: x^2 = 4, x \text{ is odd}\}$$
The only solution to $x^2 = 4$ are $x = -2$ or $x = 2$, neither of which is odd. Hence there are $no$ elements in $P$; that is, $\;P = \varnothing$.
$$Q= \{x: x^2 = 9, x \text{ is even}\}$$
The only solutions to $x^2 = 9$ are $x = -3$ or $x = 3$, neither of which is even. Hence, there are no elements in $Q$; that is, $\;Q = \varnothing$.
$$R = \{x: x^2 = 9, 2x =4\}$$
$x = 2$ is the only solution to $2x = 4$, but $x = 2$ is not a solution to $x^2 = 9$, (and neither $x = 3$ nor $x = -3$ is a solution to $2x = 4$). Hence, there are no elements in $R$; that is, $\;R = \varnothing$.
NOTE: As an aside, regarding notation - sometimes instead of a colon :preceding the defining characteristics of a given element, you'll see | in place of the colon. E.g., $$P = \{x: x^2 = 4, x \text{ is odd}\}\iff \{x\mid x^2 = 4, x \text{ is odd}\}$$
A Null Set is a set with no elements. While the author of your book uses the notation $\emptyset$, I prefer to use $\{\},$ to emphasize, that the set contains nothing. The example sets $P,\ Q$ and $R$ are all null sets, because there is no $x$, that can satisfy the condition of being included in the set.
The null set, also known as empty set, is the set containing no elements, denoted by $\emptyset$ or {}
The zero set of of a real-valued function f : X โ R is the subset of X on which f(x) = 0.
One could say that "zero set" is both defined and declared, but "null set" and "empty set" are only defined.