Mathematical Writing - Vivaldi Franco 2014
Some Advanced Terms - Families of Sets
Essential Dictionary I
We develop some advanced terminology and notation on sets. The content of this section is not essential for the rest of the book.
2.3.1 Families of Sets
The power set of a set is the set of all subsets of . Thus if , then
If has elements, then has elements. Indeed to construct a subset of we consider each element of and we decide whether to include it or leave it out, giving binary choices. (A rigorous proof requires induction, see Exercise 8.4.)
A partition of a set is a collection of pairwise disjoint non-empty subsets of whose union is . So a partition of is a subset of . For instance, the set is a partition of , and the even and odd integers form a partition of . A partition may be described as a decomposition of a set into classes.
We write some phrases using these terms. Consider carefully the distinction between definite and indefinite articles (see Sect. 2.1.4 and Exercise 2.9).
· The power set of a finite set.
· The power set of a power set.
· A partition of a power set.
· The set of all partitions of a set.
· A set of partitions of the natural numbers.
· A finite partition of an infinite set.
Now some sentences:
· Let us partition our interval into finitely many equal sub-intervals.
· The plane may be partitioned into concentric annuli.
· There is no finite partition of a triangle into squares.