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 $\emptyset \subseteq X$? 
 A: $(1)$ 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 $\varnothing\subseteq X$
A: The relation $\subseteq$ is just like the relation $\le$. It is not wrong to say that $x \le x$, i.e., "$x$ is less than or equal to $x$", although it's also true that $x = x$. The first is a weaker statement than the second, in that it conveys less information. But it's certainly not false in any way. Look at it another way: $\le$ [is less than or equal to] means the same thing as $\not >$ [is not greater than]. Is it true that $x$ is not greater than $x$? Yes. Then it is true that $x$ is less than or equal to $x$.
Similarly, $\subseteq$ [is a subset of] is equivalent to saying "does not contain elements not in". For example, $X \subseteq Y$ means "$X$ does contain any element that is not in $Y$". Is it true that $X$ does not contain any element not in $X$? Yes, it is always true. So $X \subseteq X$.
Now, let's apply this definition to the statement $\phi \subseteq X$. We know that $\phi = \{\}$, and let $X = \{1, 2, 3\}$, as in your example. Now, is it true that $\phi$ does not contain any element not in $X$? Yes! So $\phi \subseteq X$.
In more formal notation:


*

*$X \subseteq Y \Leftrightarrow (\forall x \in X, x \in Y) \Leftrightarrow (\not\exists x \in X, x \notin Y)$

*$(\forall x \in X, x \in X) \Rightarrow X \subseteq X$

*$\not\exists x \in \phi \Rightarrow (\not\exists x \in \phi, x \notin X) \Rightarrow \phi \subseteq X$

A: A very good principle when starting out in any new area of maths: get really, really clear about key definitions. Don't just make a stab at guessing (or go by the perhaps misleading everyday terminology used in labelling, or by an informal gloss on the definition): look carefully at how key notions are rigorously defined.
Now, what does $X \subseteq Y$ mean, officially?

$X \subseteq Y$ holds, by definition, if and only if, for any $x$, if $x \in X$ then $x \in Y$.

Informally, yes, you might be tempted to gloss $X \subseteq Y$  as e.g. "$X$ is a part of $Y$"; that can be useful, but as you are finding, it can also be misleading. So forget for a moment the rough informal gloss and concentrate on the official definition.
Applied to your two cases, we have first, by definition

$X \subseteq X$ if and only if, for any $x$, if $x \in X$ then $x \in X$.

Well now ask: Is it the case that for any $x$, if $x \in X$ then $x \in X$? And so what does that tell you about $X \subseteq X$?
Second, we have by definition 

$\emptyset \subseteq X$ if and only if, for any $x$, if $x \in \emptyset$ then $x \in X$.

Well now ask: Is it the case that for any $x$, if $x \in \emptyset$ then $x \in X$? Recall that $x \in \emptyset$ is always false: what happens to quantified conditionals where the antecedent is always false? And so what does that tell you about $\emptyset \subseteq X$?
A: Both questions really boil down to how mathematicians use logic.
1.) $X \subseteq X$ means that for all elements of $X$ they are elements of $X$.
2.) $\emptyset \subseteq X$ means that for all elements of $\emptyset$ they are elements of $X$.
To see why 1.) is true, consider an element $x$ in the set $X$, well then we know that $x$ is in $X$ (!) and so we have shown that all elements from $X$ are also elements of $X$.
To see why 2.) is true, we must show that given an element of $\emptyset$ it must belong to $X$. But, there are no elements of $\emptyset$ and so the implication is (trivially) satisfied. 
If you've studied truth tables then this corresponds to the case for "$P \Rightarrow Q$" where "P" is false.
A: $X$: "Hello set $Y$, is it so that any element of you is also an element of me?"
$Y$: "Yes."
$X$: "Well let us practicize a notation that expresses this fact: $Y\subseteq X$."

If $X$ would have asked the same question to himself then the answer would have been 'yes' too, right? So in the suggested notation: $X\subseteq X$.

Concerning 2): 
$X$: "Hello set $\emptyset$, is it so that any element of you is also an element of me?"
$\emptyset$: "Well, I have no elements that are not. Even stronger, I have no elements at all. So again the answer is yes"
$X$: "Well let us practicize a notation that expresses this fact: $\emptyset\subseteq X$."
