A partition of a set \(A\) is a collection of disjoint subsets whose union is all of \(A.\)

In other words, let \(\{A_i : i \in I\}\) be a collection of sets for some index set \(I\) such that

• the sets are disjoint: \(A_i \cap A_j = \emptyset\) if \(i \neq j,\) and
• the union of the sets is \(A:\) \(\bigcup_{i \in I} A_i = A.\)

Let's look at some very simple examples. Let \(A = \{1, 2, 3, 4, 5\}.\)

One partition of \(A\) is \(A_1 = \{1, 2\}\) and \(A_2 = \{3, 4, 5\}.\)

• \(A_1 \cap A_2 = \emptyset\), and
• \(A_1 \cup A_2 = A.\)

Another partition of \(A\) is \(A_1 = \{1\},\) \(A_2 = \{2, 3\},\) and \(A_3 = \{4, 5\}.\) These sets are all contained in \(A,\) they are disjoint, and their union is \(A.\)

The sets \(A_1 = \{1, 2\}\) and \(A_2 = \{2, 3, 4, 5\}\) are not a partition of \(A\) because they are not disjoint. \(A_1 \cap A_2 = \{2\}.\)

The sets \(A_1 = \{1, 2\}\) and \(A_2 = \{4, 5\}\) are not a partition of \(A\) because their union is not all of \(A\). \(A_1 \cup A_2 = \{1, 2, 4, 5\}.\)

Example of a partition with pictures:

Let \(A\) be represented by the red circle.

A partition of \(A\) is a way to divide \(A\) into separate parts. This picture shows \(A\) partitioned into sets \(A_1, A_2, A_3, A_4,\) and \(A_5.\)