Definition 17 (Partition). A partition of X is a collection ℱ of subsets of X such that (i) any two distinct elements of ℱ are disjoint, and (ii) ⋃ ℱ = X.
19Partitions
A partition of a set X is a collection ℱ of subsets of X satisfying two conditions:
The idea is that a partition cuts X into pieces, with nothing left out and no overlaps.
For example, {{7, 8}, {9}, {10, 11}} is a partition of {7, 8, 9, 10, 11}. But {{7, 8}, {8, 9}} is not a partition of {7, 8, 9}, because the two pieces overlap; and {{7, 8}, {9}} is not a partition of {7, 8, 9, 10}, because 10 is left out.
A picture of a partition of X into four pieces:
Proposition 47 (each element lies in exactly one cell). If ℱ is a partition of X, then every element of X belongs to exactly one element of ℱ.
Let x be an arbitrary element of X. We want to show that x belongs to exactly one element of ℱ.
First, we show that x belongs to at least one element of ℱ. By part (ii) of the definition of a partition, we have x ∈ ⋃ ℱ, hence by definition of union there is some Y ∈ ℱ such that x ∈ Y.
Next, we show that x belongs to at most one element of ℱ. Suppose for contradiction that x belongs to two distinct elements of ℱ, which we can denote by Y and Z. Then, by the definition of pairwise intersection, we have x ∈ Y ∩ Z. Hence Y ∩ Z ≠ ∅, contradicting condition (i), which implies that Y and Z are disjoint.
The decompositions in Section 17 were partitions into two pieces.