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?
$\endgroup$5 Answers
$\begingroup$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.
$\endgroup$ 5 $\begingroup$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 $P_1\cup P_2\cup P_3=X$.
$\endgroup$ $\begingroup$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}
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$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}.
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