1.5 Partitions of Sets
Pairwise Disjoint: Let \(A\) be a set. A collection \(S\) of subsets of \(A\) is pairwise disjoint if every two distinct sets in \(S\) are disjoint.
Partition: Let \(A\) be a set. If \(S\) is a collection of non-empty subsets of \(A\) and
- \(S\) is pairwise disjoint and
- the union of the subsets in \(S\) is \(A\)
then we say \(S\) is a partition of \(A\).
This can be useful if, for example, we're trying to prove a theorem for the integers \(Z\). If we can prove the theorem for each partition, for example the positives, negatives, and zero separately, then we prove for the whole set \(Z\).