Partition of a set, definition not clear From wikipedia:

Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:
  
  
*
  
*The union of the elements of P is equal to X. (The elements of P are said to cover X.)
  
*
  The intersection of any two distinct elements of P is empty. (We say the elements of P are pairwise disjoint.)
  

I clearly understand that the intersection between partition is empty (point 2), but how can the union of a partition can be the all elements in the set?
If it is a partition, shouldnt they be just a part?
I imagine a set divided in 3 and the elements in the first part are not all the elements of the second part.
How do you explain this?
 A: The union of all parts gives you the whole set. So if you partition a set $X$ in three parts $P_1$, $P_2$, $P_3$, then $P_1\cup P_2\cup P_3=X$.
A: The examples will help. Examples of partitions of $ \{1,2,3\} $ are
\begin{equation}
\{1\}, \{2\}, \{3\}
\end{equation}
\begin{equation}
\{1,2\},\{3\}
\end{equation}
\begin{equation}
\{1\},\{2,3\}
\end{equation}
\begin{equation}
\{1,2,3\}
\end{equation}
\begin{equation}
\{2\},\{1,3\}
\end{equation}
A: The idea of a partition is that you take a whole (the set $X$) and you divide it to parts.
Now if I cut off an apple into slices (and one core) I have several pairwise disjoint parts of the apple, but if I reassemble the parts I get a whole apple again.
Similarly we require this from a partition of a set. We want that the union of all the parts give us the entire set we partitioned.
A: I believe your confusion regarding the definition of a partition, P, of a set X may stem from conflating the elements of X with the elements of P. The elements of a partition are non-empty subsets of X. For P to be a partition of X it's elements (subsets of X) must be disjoint and cover all of X.
If you keep in mind that the elements of P are non-empty subsets of X, things should fall into place.
A: A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set.
For example, one possible partition of {1, 2, 3, 4, 5, 6} is {1, 3}, {2}, {4, 5, 6}.
