elementary set theory - What is a null set? - Mathematics Stack Exchange
Why do we consider the null set to be a subset of all sets? Could we define it otherwise?
How can I differentiate between "zero" and "null"?
What is the difference between No Solution and Null Solution?
What is a null set example?
How do you define a null set?
What is a null set called?
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The distinction between the empty set 
and the number 
is similar to that between NULL and ZERO. For example, the set of real solutions (or informally "the solution") to 
is 
, but the solution to 
is 
.
In my mind there is no need for a concept like NULL in mathematics if you think of NULL as in NULL-pointers.
NULL in this sense is a technical necessity because you cannot un-define a variable: Once a variable has been assigned a value, a certain bit of memory is reserved for this variable and this memory is marked as re-usable only if the variable goes out of scope (simplified speaking).
You cannot say "The variable with this name doesn't exist anymore." without letting it go out of scope, because that would make language interpretation much more complicated without many benefits. Therefore, to indicate that the value of the variable has no meaning, one uses NULL.
What NULL stands for in the end depends upon the programming language: In some it is a special keyword, but in some it is also just a different name for the integer 
.
You can assign an arbitrary value to NULL in mathematics as mentioned in the other replies (
, 
, etc.) but as mathematics has nothing to do with memory allocation there is really no need for such a thing as NULL.
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.
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.