Is {∅} a subset of the power set of X, for every set X? I am reading the book "A Transition to Advanced Mathematics" 2014 edition, written by Smith, Eggen and St. Andre.
For every set X:
∅ ⊆  X
∅ ∈  power set of X
∅ ⊆  power set of X
{∅} ⊆  power set of X
Is {∅} ⊆ power set of X wrong?
I am an undergraduate student. Please help me!
The definition of power set of X is the set whose elements are subsets of X. (page 90 of the book)
I have this doubt because I remember that I read somewhere that ∅ ≠ {∅} because a set with the empty set is not empty.
Thanks for your answers!
What about this question:
If the definition of power set of X is the set whose elements are subsets of X, and {∅} is not a subset of X. How {∅} can be a subset of the power set of X? Is not that an contradiction?
 A: Recall that $A\subseteq B$ if whenever $a\in A$, then we have that $a\in B$ as well.
Since $\varnothing\in\mathcal P(X)$, it means that every element of $\{\varnothing\}$ is also an element of $\mathcal P(X)$, so $\{\varnothing\}\subseteq\mathcal P(X)$ is indeed correct.
A: The empty set $\emptyset$ is a subset of every set $X$. Hence $\emptyset$ is a member of the power set of $X$, $P(X)$. Therefore $\{\emptyset\}$ is a subset of $P(X)$.
A: Let's do this with an example.
Here's a set: $\{0,1\}$. Its power set ($\mathcal{P}$) is: $\{ \emptyset,\{0\},\{1\},\{0,1\} \}$. That's four elements. We can check that we calculated the power set correctly because we know that for a set $A$ with size $|A|$, the size of the power set is always $2^{|A|}$, and $2^2$ is $4$.
Here's another example: take the set $\{2,3,4\}$. It is clear that $\{2,3\}$ is a subset. And so $\{2\}$ is a subset as well.
With this example, we can see that $\{\emptyset\} \subseteq \mathcal{P}$ holds. Clearly $\{\emptyset,\{0\}\}$ is a subset of $\mathcal{P}$, so $\{\emptyset\}$ must be, too, even if may look unusual at a first glance.
A: One of those statements doesn't belong with the others. "∅ ⊆  power set of X" is correct, but irrelevant. I think that's what's confusing you.
