A set $X$ containing 2 elements that are also subsets of $X$ The question is pretty much summed up in the title.
I am taking an introductory Discrete Mathematics course and our teachers asked us to find a set $X$ containing only 2 elements where those elements are also subsets of X.
The teacher also said that there is a unique answer to this question, meaning there can't be more than 1 correct answer.
$X = \{a, b\} \ : a \subset X, b \subset X$
An option I thought of is $X = \{\phi, X∁ \}$ (the 2nd element is the complement of X)
Is this correct? If not, what would be the correct answer?
 A: Hint. 
Here is a set with one element that happens to be a subset of itself:
$$
A = \{ \phi \} .
$$
Your two element set will have to have $A$ as one of its elements ...
(I think this is a pretty tricky question for beginning discrete mathematics.)
A: $X=\{\emptyset, \{\emptyset\}\}$ will do.
$\emptyset \in X$ and always $\emptyset \subseteq X$.
$\{\emptyset\} \in X$ and $\{\emptyset\} \subseteq X$ as the only element of $\{\emptyset\}$ is indeed also in $X$.
Such sets are called transitive by the way. (The terminology comes from the observation that $x$ is transitive iff $\forall y,z: (y \in z) \land (z \in x) \implies y \in x$) 
A: How would we go about this?
Well first $X=\{a, b\}$ and $a\subseteq X$ means that $a=\emptyset$ or $a=\{a\}$ or $a=\{b\}$ or $a=X$
And similar options apply for $b$. The second and fourth of these are impossible as a set cannot be a member of itself.
Let's take the first $a=\emptyset$. Then $b\neq a$ so the only possibility is $b=\{\emptyset\}$ and this in fact works.
The alternative is to start with $a=\{b\}$ in which case we are not allowed $b=\{a\}$ and must have $b=\emptyset$, and we get the same solution.
A: You define $X$ in terms of itself, which would be considered logically unsound. The complement is also only defined with regards to some ambient set, but no ambient set is defined here.
Here is the answer:
$$X=\{\varnothing,\{\varnothing\}\}$$
Remember that the empty set is a subset of every set.
