Some Advanced Terms - Families of Sets - Essential Dictionary I

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 $$\mathbf {P}(A)$$ of a set $$A$$ is the set of all subsets of $$A$$. Thus if $$A=\{1,2,3\}$$, then

$$ \mathbf {P}(A)=\left\{ \{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\right\} . $$

If $$A$$ has $$n$$ elements, then $$\mathbf {P}(A)$$ has $$2^n$$ elements. Indeed to construct a subset of $$A$$ we consider each element of $$A$$ and we decide whether to include it or leave it out, giving $$n$$ binary choices. (A rigorous proof requires induction, see Exercise 8.4.)

A partition of a set $$A$$ is a collection of pairwise disjoint non-empty subsets of $$A$$ whose union is $$A$$. So a partition of $$A$$ is a subset of $$\mathbf {P}(A)$$. For instance, the set $$\left\{ \{2\},\{1,3\}\right\} $$ is a partition of $$\{1,2,3\}$$, and the even and odd integers form a partition of $$\mathbb {Z}$$. 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.