elementary set theory - What is a null set? - Mathematics Stack Exchange
Define null set, give example.
Why do we consider the null set to be a subset of all sets? Could we define it otherwise?
ELI5: What is a null set?
What is a null set called?
How do you define a null set?
Is รธ an empty set?
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Perhaps what you find confusing is the use of set-builder notation to define 
: 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 
, 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 
is the set of all x such that x satisfies (condition 1) AND x satisfies (condition 2) AND ... AND x satisfies (condition n).
The only solution to 
are 
or 
, neither of which is odd. Hence there are 
elements in 
; that is, 
.
$$Q= \{x: x^2 = 9, x \text{ is even}\}$$
The only solutions to 
are 
or 
, neither of which is even. Hence, there are no elements in 
; that is, 
.
is the only solution to 
, but 
is not a solution to 
, (and neither 
nor 
is a solution to 
). Hence, there are no elements in 
; that is, 
.
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., 
A Null Set is a set with no elements. While the author of your book uses the notation 
, I prefer to use 
to emphasize, that the set contains nothing. The example sets 
and 
are all null sets, because there is no 
, 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:
โThe 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 ร or {}.โ
โThe reasoning for considering the null set as a subset of S is quite interesting: If the null set is not a subset of S (ร โ S), then ร must contain at least one element ๐ฅ such that ๐ฅ โ S. But since by definition the null set has no element whatsoever, we cannot say that ร โ S; hence the null set is a subset of Sโ
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 ๐ฅ such that ๐ฅ โ S AND no one element ๐ฅ such that ๐ฅ โ S?
My intuitive thinking:
If I have an apple, an orange, and a kiwi, I usually donโt also go around thinking that I also have a โno fruitโ. Feels wrong to claim that โno elementโ 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.