TITLE: Can we say that an element is not a subset of a set?
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: 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 !