TITLE: How can an ordered pair be expressed as a set?
QUESTION [43 upvotes]: My book says 
\begin{equation}
(a,b)=\{\{a\},\{a,b\}\}
\end{equation}
I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made any sense to me.
For example, Definition of an Ordered Pair
Based on how my ignorant brain is viewing this, I don't see why the definition could not be.
\begin{equation}
(a,b)=\{\{a\},\{b\},\{a,b\}\}
\end{equation}
aka the power set. What is the significance of the {a} in that definition? Please keep things simple if possible. Normally definitions have a valid and clear reason for being defined that way.

Clarification
First, I understand what an ordered pair is. I just don't see how the set notation says that. 
Second, 
\begin{equation}
(a,b) = \{\{a\},\{a,b\}\}=\{\{a,b\},\{a\}\}
\end{equation}
Sets don't preserve order, but ordered pairs do. How does the third part of the equality apply to the definition?
Third, another issue with the notation that I have starts with the Product Property of Sets
\begin{equation}
\text{Let $X$ and $Y$ be sets} :\ X=\{a,b,c\}\text{ and }Y=\{a,d,e\}.
\end{equation}
\begin{equation}
\text{Then }X \times Y = \{(a,a),(a,d),(a,e),(b,a),(b,d),(b,e),\dots,(c,e)\}
\end{equation}
If we look at the first ordered pair and our given definition we have
\begin{equation}
(a,a)=\{\{a\},\{a,a\}\}
\end{equation} 
How can this be so, you can't have duplicates in sets? I guess what I am looking for in an answer, is not a proof or a definition of ordered pairs, but rather something like, "This notation says what it says because...". Except for the second to last point I get the terminology, I just don't get the connection between the two different uses of notation.

REPLY [2 votes]: You can also think a pair is a set of two elements where the order is specified with identifying the first element.