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., 
ELI5: What is a null set?
Why do we consider the null set to be a subset of all sets? Could we define it otherwise?
Empty Set and NULL Set part of a define set?
X is a subset of Y means every element of X is an element of Y.
And in fact, every element of C is an element of B. If you don't believe me, can you find me an element of C that isn't an element of B? Since there is no such element, every element of C is an element of B, and so C is a subset of B.
Now B is not a subset of C, because it is not the case that every element of B is an element of C. For example, the element 2 is an element of B but not an element of C. So B is not a subset of C.
More on reddit.comelementary set theory - Null Sets $\{\{\emptyset\}\} \subset\{\emptyset, \{\emptyset\}\}$ - Mathematics Stack Exchange
What is a null set called?
How do you define a null set?
Is ø 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.,
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:
“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.
U = {N|N=All real number} A = {1,3,5,7,9} B = {2,4,6,8} C = {}
How is C a subset of U, A, and B but U, A, B is not a subset of C?
Please show me mathematically how a NULL or empty set is contained within a define set. Or use any deductive reasoning that makes sense.
All I got was this - You have a used car sale lot that sells car (U). In the used car sale lot, they have Toyotas (A). They also have Fords (B). They only have Toyotas and Fords. That is it. Nothing else. Because it is defined as having Toyotas and Fords but not empty, how can any empty set be a part of it? If we were to assume that an empty set can be a part of the dealer because it they can run out of cars, what is stopping someone to make the assumption that they can have a Chevy in there too at one point in time, since they are just a used car sale lot.
Am I making any sense? Please help me understand.
EDIT: I guess I look at things differently then the rest of the world... I always thought that a set define a group value. Key word, define. So when you have a set of, say P{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto}, you have a set called P defining the planets within the Sol solar system. Since it did not define an emptiness, how can an empty set be contain within it? sigh In all honestly, I probably am thinking too deeply at this.
X is a subset of Y means every element of X is an element of Y.
And in fact, every element of C is an element of B. If you don't believe me, can you find me an element of C that isn't an element of B? Since there is no such element, every element of C is an element of B, and so C is a subset of B.
Now B is not a subset of C, because it is not the case that every element of B is an element of C. For example, the element 2 is an element of B but not an element of C. So B is not a subset of C.
null set and empty set are different things. a null set is usually a set of measure zero. the empty set if a subset of every set because all elements of the empty set (there exist none) are also in A (and B and U). meanwhile none of the elements of A, B or U are in the empty set, because it is empty. so A B and U are not subsets of C.
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