Problem 43 - Discrete Mathematics with Applications

Question

Determine if each is a partition of the set \(\\{a, \ldots, z, 0, \ldots, 9\\}.\) $$\\{\\{a, \ldots, z\\},\\{0, \ldots, 9\\}, \emptyset\\}$$

Step-by-Step Solution

Answer

The given collection is not a partition of the set \( \\{a, \ldots, z, 0, \dots, 9\\} \) because it includes the empty set (\( \emptyset\)) as one of its subsets, which is not allowed in a partition.

Step 1: 1. Verify that no subset in the collection is empty

Check if any of the given subsets is the empty set (\( \emptyset \)). In the given collection, we have the empty set as one of the subsets, which is not allowed in a partition. Hence, this condition is not satisfied.

Step 2: 2. Check if each element in the original set belongs to exactly one subset in the collection

Since we have already identified that the empty set is a subset in the collection, there is no need to proceed with this step, as the first condition for being a partition was not satisfied.

Step 3: 3. Conclusion

Since one of the conditions for being a partition is not satisfied (empty set is included in the collection), the given collection is not a partition of the set \( \\{a, \ldots, z, 0, \dots, 9\\} \).

Key Concepts

Partition of a Set

When discussing set theory, a "partition of a set" is a way of breaking down a larger set into smaller, non-overlapping subsets such that every element of the original set belongs to exactly one of these subsets. To form a valid partition, a few rules have to be followed:

• Each subset in the collection must be non-empty. A partition cannot include the empty set.
• The collection of subsets must be mutually exclusive, meaning no overlap is allowed between the subsets.
• All elements of the original set must be included in one and only one of the subsets.

The exercise you encountered required checking if a collection containing an empty set is a partition. Since one of the subsets in the collection was empty, it failed to meet the criterion of non-emptiness. Therefore, the collection is not a valid partition of the set \(\{a, \ldots, z, 0, \ldots, 9\}\). Understanding these rules can help you determine whether any collection of subsets forms a partition.

Subsets

A subset is essentially a set formed by selecting some (or all) elements of another set without changing their order. Every set is a subset of itself, and a subset may contain no elements at all, also known as the "empty set." Here are the key points about subsets:

• If every element of set A is also an element of set B, then A is referred to as a subset of B.
• A subset that includes some but not all elements of its original set is known as a "proper subset."
• If a set A is a subset of another set B, we write this as \(A \subseteq B\).

In the context of the exercise, understanding subsets helps clarify why including an empty set within a collection makes it an invalid partition. Subsets must contain members of the original set they derive from; thus, they cannot be completely empty when forming a partition.

Empty Set

An empty set, denoted as \(\emptyset\), is a fundamental concept in set theory. It is the unique set that contains no elements whatsoever. Despite its lack of elements, it still qualifies as a set. Here are essential features of an empty set:

• The empty set is a subset of any set, including itself.
• It is symbolized by \(\{\}\) or \(\emptyset\).
• Even though it doesn't contain any elements, it is crucial for operations like difference or in defining subsets.

In the exercise, the ordering principle of partitions prohibits the inclusion of the empty set. This is because a partition must cover the entire original set without omissions. Including an empty set means there are elements in the original set that do not belong to any subset, violating the criteria for a partition.