Partitioning a Set

Let $ A $ be a set, and let $ A_1, A_2, ..., A_n $ be subsets of $ A $. These subsets form a partition of $ A $ if they satisfy the following three fundamental conditions:

Non-empty subsets: Each subset $ A_1, A_2, ..., A_n $ contains at least one element. \[ A_i \neq \emptyset, \quad \forall i = 1, 2, ..., n \]
Pairwise disjoint: No two distinct subsets share any elements. \[ A_i \cap A_j = \emptyset, \quad \forall i \neq j \]
Complete coverage: The union of all subsets reconstructs the original set $ A $, meaning every element of $ A $ belongs to exactly one subset. \[ A = \bigcup_{i=1}^{n} A_i = A_1 \cup A_2 \cup ... \cup A_n \]

In simple terms, a partition completely breaks down the set $ A $ into distinct, non-overlapping subsets, ensuring that no elements are left out.

Let’s explore this concept with some concrete examples.

Examples of Set Partitions

Example 1

Consider the finite set $ A = \{1, 2, 3\} $.

One possible way to partition this set is:

$ A_1 = \{1\} $ and $ A_2 = \{2, 3\} $.

We can verify that this satisfies all the conditions for a partition:

Each subset is non-empty: $$ A_1, A_2 \ne \emptyset $$ The subsets are disjoint (i.e., they have no elements in common): $$ A_1 \cap A_2 = \emptyset $$ Their union reconstructs the original set: $$ A_1 \cup A_2 = A $$

Similarly, other valid partitions include:

$ B_1 = \{2\} $ and $ B_2 = \{1, 3\} $,

$ C_1 = \{3\} $ and $ C_2 = \{1, 2\} $.

In every case, the subsets are non-empty, mutually exclusive, and together they completely cover $ A $.

Example 2

Now, let’s consider an infinite set: the set of natural numbers $ \mathbb{N} $. We can partition it into two subsets:

The set of even numbers: $ \{0, 2, 4, 6, 8, ...\} $, and the set of odd numbers: $ \{1, 3, 5, 7, 9, ...\} $.

$$ A = \{0, 2, 4, 6, 8, ...\} $$

$$ B = \{1, 3, 5, 7, 9, ...\} $$

These two subsets form a valid partition of $ \mathbb{N} $ because:

Neither subset is empty.
They do not overlap (a number is either even or odd, never both).
Their union covers all natural numbers: $$ A \cup B = \mathbb{N} $$

This demonstrates that partitions are not limited to finite sets—they can also be applied to infinite ones.

Example 3

Now, let’s look at a non-numerical example using a set of words:

$$ S = \{"cat", "dog", "fish", "elephant"\} $$

We can partition this set based on a specific criterion—say, the number of letters in each word:

$ S_1 = \{"cat", "dog"\} $ (3 letters)
$ S_2 = \{"fish"\} $ (4 letters)
$ S_3 = \{"elephant"\} $ (8 letters)

Once again, this satisfies the partition conditions:

Each subset is non-empty.
There is no overlap—each word belongs to exactly one subset.
The union of these subsets reconstructs the original set: $$ S = S_1 \cup S_2 \cup S_3 $$

Being able to systematically divide a set based on clear criteria is a fundamental skill in mathematics, science, and countless real-world applications.

So next time you categorize objects, groups, or data into distinct, non-overlapping categories, remember: you’re intuitively applying the mathematical principle of partitioning!