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?

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 $P(X)$.
