Can we say that an element is not a subset of a set? I am creating a multiple choices question for students that should be as follows:
Which of the following is true?
1) $8\in \mathbb{Z^{+}}$
2) $8\notin \mathbb{Z^{+}}$
3) $8\subset \mathbb{Z^{+}}$
4) $8 \not\subset \mathbb{Z^{+}}$
Should choices Number (1) and (4) be considered as valid answers?
My understanding is that we can't deal with an element with either $\subset$ or $\not\subset$.
Please give me references supporting your answer if there.
 A: If you want to initiate a class discussion on what "8" might mean to a mathematician and whether $A \not \subset B$ means the same as $\neg (A \subset B)$ when A or B is not even a set (or is everything in mathematics built from a set anyway ?), then this is a great way to start.
If you wish to leave those topics until another occasion then I suggest you change the question !
A: The easy answer is that only option 1 (and possibly 4) is true, that's enough for naive set theory. 
Note that even if $8$ is not a set the construct $8\subset\mathbb Z^+$ is a well-formed statement (the only problem is that it's false, only sets can be subsets). The statement 4 is true in the sense that it's the negation of $3$, but in naive set theory one could also use the definition that such statements being false as $8$ is not a set.
A: Yes, an element of a set can be a subset of that set. 
Think of $\varnothing\in\{\varnothing\}$ and $\varnothing\subset\{\varnothing\}$, where $\varnothing$ denotes the empty set.
The answer to your multiple choice question depends on how $\mathbb Z_+$ and $8$ are defined.
A possible construction exists with $\mathbb Z_+=\{1,2,3,\dots,8,\dots\}$ and $8=\{1,2,3,4,5,6,7\}$.
In that case $8\in\mathbb Z_+$ and $8\subset\mathbb Z_+$.
A: For me the option 1 is valid, for options 3 and 4 proper notation of set isn't used. If we require to represent a set with elements then the elements should be surrounded by curly braces . Hence 8 doesn't represents a set therefore 3 and 4 options are invalid.
This wikipedia post suggests the notation used for set elements
