Definition – Partition – Math-reference.net

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Definition – Partition of a subset of $\mathbb{R}$

Let $S=[a,b]$ be a closed subset of $\mathbb{R}$ . We refer to a set

$X = \{x_1, ..., x_n\}$

As a partition of the set $S$ .

This is of course a more specific example of the next definition, as we are breaking down the interval $[a,b]$ into pairwise disjoint subsets. However, remember that the intervals making up the partition will be closed on one side and open on the other, as otherwise the sets would not be pairwise disjoint, or some point of $S$ would not be included in the partition.

The largest interval in $X$ is called the mesh of the partition.

Definition – Partition of any set

Let $S$ be a set. Let $F$ be a family of subsets of $S$ such that for all pairs of sets in $F$ , say $A\in F$ and $B\in F$ ,

$A\cap B = \emptyset$ .

If $F$ is such a set, then we say that $F$ is a partition of $S$ , and that $F$ partitions $S$ .