subsets and sets I'm not so bad in math however I want to understand math the way mathematicians do, so I picked up this book How to Think Like a Mathematician. The book seems the book that I'm looking for. Definitely, it gonna be very long dark tunnel to me until I see the light. Any ways, he started with sets and I came across the following 
1- For any set $X$, we have $X \subseteq X$.
2- For any set $X$, we have $\emptyset \subseteq X$.
I couldn't imagine  what do the two aforementioned statements imply. For the first one, how come a set is part of or equal itself? I can imagine the equality here but not the partition. The second statement, how come a non-empty set has an empty set? For example, let $ X = \{1, 2, 3\}$, now why $\emptyset \subseteq X$? 

$(1)$ Lets consider something that may seem more familiar. The symbol "$\le$" means less than on or equal to, so if we see $a\le b$, one of the following criteria hold:
$(i)\,a\lt b$ ($a$ is strictly less than $b$), or
$(ii)\, a=b$.
Now it is clear that $a\le a$ because $(ii)$ holds.
Now we will work with subsets, $Y\subseteq X$ if:
$(i)$ every element of $Y$ is an element of $X$ (but $Y\ne X$), or
$(ii)$ $Y=X$
Now it is clear that $X\subseteq X$, because $(ii)$ holds.
$(2)$ the empty set $\varnothing$ is the set consisting of no elements. It is true that for any set $X$ we have $\varnothing\subseteq X$. So let us look at this constructively, as this a good way to see how things work.
How would one create a subset of $X$? One way would be to just remove elements of $X$, then what is left over would satisfy $(i)$, and thus be a subset.
Now remove all the elements of $X$ from $X$, what are you left with? A set consisting of nothing, I.e the eempty set, but still $(i)$ holds, so we see that $\varnothing\subseteq X$
