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Step 1: Understand the definitions provided for a partition of a set u005C( A u005C).u000Au000AStep 2: Analyze D

We mentioned that there are three ways that a collection 𝒮...

We mentioned that there are three ways that a collection $\mathcal{S}$ of subsets of a nonempty set $A$ is defined to be a partition of $A$. Definition 1 The collection $\mathcal{S}$ consists of pairwise disjoint nonempty subsets of $A$ and every element of $A$ belongs to a subset in $\mathcal{S}$. Definition 2 The collection $\mathcal{S}$ consists of nonempty subsets of $A$ and every element of $A$ belongs to exactly one subset in $\mathcal{S}$. Definition 3 The collection $\mathcal{S}$ consists of subsets of $A$ satisfying the three properties (1) every subset in $S$ is nonempty, (2) every two subsets of $A$ are equal or disjoint and (3) the union of all subsets in $S$ is $A$. (a) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 1 satisfies Definition 2. (b) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 2 satisfies Definition 3. (c) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 3 satisfies Definition 1.

We mentioned that there are three ways that a collection $\mathcal{S}$ of subsets of a nonempty set $A$ is defined to be a partition of $A$.
Definition 1 The collection $\mathcal{S}$ consists of pairwise disjoint nonempty subsets of $A$ and every element of $A$ belongs to a subset in $\mathcal{S}$.
Definition 2 The collection $\mathcal{S}$ consists of nonempty subsets of $A$ and every element of $A$ belongs to exactly one subset in $\mathcal{S}$.
Definition 3 The collection $\mathcal{S}$ consists of subsets of $A$ satisfying the three properties (1) every subset in $S$ is nonempty, (2) every two subsets of $A$ are equal or disjoint and (3) the union of all subsets in $S$ is $A$.
(a) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 1 satisfies Definition 2.
(b) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 2 satisfies Definition 3.
(c) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 3 satisfies Definition 1.

Key Concepts

Partition of a Set

A partition of a set is a division of the set into distinct, nonempty subsets such that every element of the set belongs to exactly one of these subsets. This concept is central in set theory as it formalizes the idea of breaking a set into mutually exclusive and collectively exhaustive parts.

Pairwise Disjoint Subsets

Pairwise disjoint subsets are subsets in which any two different subsets have no elements in common. In the context of partitions, this property ensures that each element of the set is allocated to only one subset, thereby preventing overlaps between the subsets.

Covering Property

The covering property refers to the requirement that the union of all subsets in the collection equals the original set. This guarantees that every element of the set is included in at least one subset, thereby ensuring that the partition is complete.

Uniqueness of Membership

Uniqueness of membership is the idea that every element of the set belongs to one and only one subset of the partition. This property refines the covering property by excluding the possibility of an element falling into more than one subset, which is fundamental for the structural integrity of a partition.

Equivalence of Definitions

The equivalence of definitions demonstrates that different but intuitively similar ways of defining a partition of a set ultimately describe the same mathematical structure. By showing that each set of conditions implies the others, one verifies that definitions based on pairwise disjointness, covering properties, or uniqueness of membership are all equivalent approaches to partitioning a set.

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