Problem 46 - Discrete Mathematics with Applications

Question

Determine if each is a partition of the set \(\\{a, \ldots, z, 0, \ldots, 9\\}.\) $$\\{\\{a, \ldots, u\\},\\{v, \ldots, z\\},\\{0,3\\},\\{1,2,4, \ldots, 9\\}$$

Step-by-Step Solution

Answer

The given sets form a valid partition of the set \(\\{a, \ldots, z, 0, \ldots, 9\\}\) because the union of the sets equals the original set and the intersection of any two different sets is empty.

Step 1: Check if the union of the sets equals the original set

To do this, let's first compute the union of the given sets: \(\\{a, \ldots, u\\} \cup \\{v, \ldots, z\\} \cup \\{0,3\\} \cup \\{1,2,4, \ldots, 9\\}.\) We can see the union of the given sets results in the set: \(\{a, b, c, \ldots, z, 0, 1, 2, 3, 4, \ldots, 9\}\) This set contains all the elements of the original set, so this condition is satisfied.

Step 2: Check if the intersection of any two different sets is empty

We need to check if any two sets in the partition have any common element. If the intersection is empty for every pair of distinct sets, then they're disjoint. We'll check the intersections of the following pairs of sets: 1. \(\\{a, \ldots, u\\} \cap \\{v, \ldots, z\\}\) 2. \(\\{a, \ldots, u\\} \cap \\{0,3\\}\) 3. \(\\{a, \ldots, u\\} \cap \\{1,2,4, \ldots, 9\\}\) 4. \(\\{v, \ldots, z\\} \cap \\{0,3\\}\) 5. \(\\{v, \ldots, z\\} \cap \\{1,2,4, \ldots, 9\\}\) 6. \(\\{0,3\\} \cap \\{1,2,4, \ldots, 9\\}\) For all pairs, we observe that their intersection is empty, confirming that they're disjoint. Since the union of the sets equals the original set, and the intersection of any two different sets is empty, we can conclude that the given sets form a partition of the set \(\\{a, \ldots, z, 0, \ldots, 9\\}\).

Key Concepts

Partition

In set theory, a partition of a set is a way of dividing all its elements into non-overlapping and non-empty subsets such that every element of the original set is included in exactly one of these subsets. The given exercise demonstrates an example of a partition.
To check if a collection of subsets is a partition, we must ensure the following:

• The union of all subsets equals the original set.
• Each pair of subsets is disjoint, meaning the intersection of any two subsets is empty.

In the provided problem, we examined the subsets \(\{a, \ldots, u\}\), \(\{v, \ldots, z\}\), \(\{0,3\}\), and \(\{1,2,4, \ldots, 9\}\). By combining all these subsets, we confirmed that they covered all elements from \(\{a, \ldots, z, 0, \ldots, 9\}\). Additionally, all potential intersecting pairs were checked to ensure they do not share elements. Thus, they indeed form a partition of the original set.

Union of Sets

The union of sets is a fundamental operation in set theory. It combines all the elements of the given sets into a single set. In simple terms, the union includes any element that is in any of the participating sets.If you have two sets \(A\) and \(B\), their union \(A \cup B\) consists of all elements that are in \(A\), or in \(B\), or in both.In the exercise provided, the union of all the subsets \(\{a, \ldots, u\} \cup \{v, \ldots, z\} \cup \{0,3\} \cup \{1,2,4, \ldots, 9\}\) resulted in creating the complete set \(\{a, \ldots, z, 0, \ldots, 9\}\).
This satisfies the requirement for the subsets to collectively form the entire set they partition. The simple yet powerful notion of the union allows us to gather all possible elements from different sets without missing any.

Intersection of Sets

The intersection of sets refers to the elements that are common to every set in a given collection. When considering two or more sets, we seek elements that exist in all of them at the same time.For sets \(A\) and \(B\), their intersection, denoted by \(A \cap B\), contains all elements that \(A\) and \(B\) share.
In the context of the exercise, each pair of subsets was evaluated for intersections. Our goal was to confirm that no element appeared in more than one subset since that would violate one of the partition criteria.

• \(\{a, \ldots, u\} \cap \{v, \ldots, z\} = \emptyset\)
• Similarly, \(\{0,3\} \cap \{1,2,4, \ldots, 9\} = \emptyset\)

All intersections with these characteristics confirmed that the subsets were pairwise disjoint. This detailed check ensured that each subset was wholly separate from the others, consistent with partition rules.

Disjoint Sets

Disjoint sets are sets that have no elements in common. In terms of set theory, their intersection results in an empty set, denoted by \(\emptyset\).Verifying that sets are disjoint is a crucial step when evaluating a potential partition of a set.
During the problem's step-by-step analysis, every pair of subsets was found to be disjoint, leading to empty intersections among them. This verification process includes comparisons such as:

• \(\{a, \ldots, u\}\) and \(\{v, \ldots, z\}\)
• \(\{0,3\}\) and \(\{1,2,4, \ldots, 9\}\)

Since the subsets featured in the exercise never shared elements, we could confidently affirm that each subset stands apart from the others, proving them to be disjoint sets. Thus, this confirms our constructed partition effectively divides the original set without any overlaps.