22.3.2. Trivial Partition

A partition of a set C is a collection of pairwise disjoint, non-empty subsets such that their union gives back the original set C. Each subset must have at least one element, and they must be pairwise disjoint.

A partition of a set is essentially a way to divide that set into distinct parts or subsets. The subsets must not overlap, meaning they do not share any common elements. Moreover, every element of the original set must belong to one of the subsets in the partition. For example, if we have a set of fruits {apple, banana, cherry}, we might create a partition with subsets {apple}, {banana, cherry}. Here, both subsets are non-empty, and together they contain all the original fruits.

Think of a school with different grades. Each grade (like grade 1, grade 2, etc.) can be seen as a subset of the school. They are disjoint because a student can only belong to one grade at a time, hence these subsets do not overlap. If we take all grades together, we can recreate the whole student population of the school.

A trivial partition of the set C is the set itself. For example, set C can be partitioned into just one subset, which is the entire set C.

The trivial partition is the simplest way to create a partition. When you consider the whole set as one single subset, you've created a trivial partition. For instance, if our set C contains the elements {a, b, c}, then the trivial partition can simply be {{a, b, c}}. There are no smaller subsets, and they do not intersect with each other since there is only one subset.

Imagine a complete pizza. If you consider the pizza as one whole thing without cutting it into slices, you have created a trivial partition. No individual pieces exist, just the entire pizza, similar to how the trivial partition represents the full set without subdivision.

Aside from the trivial partition, there are multiple ways to partition a set into various subsets, such as dividing set C into exactly two halves or three equal sets.

A set can be divided in numerous ways, leading to different partitions. For example, if set C contains the numbers {1, 2, 3, 4}, you could partition it into subsets like {1, 2} and {3, 4}, or {1, 3} and {2, 4}, etc. Each of these represents a valid way to partition the set and they do not overlap. There are many combinations to divide a set, indicating that partitions can vary significantly.

Consider organizing a team for a project. Depending on the objectives, you might split your team in different ways: into two groups focusing on different goals, or into three groups tackling various tasks. Each method of division is like creating a new partition of the team, just as subsets vary when partitioning a set.

An equivalence relation on set C gives rise to a partition of C into equivalence classes, which are subsets that represent the various ways elements are related to one another.

Equivalence relations group elements of a set based on a certain similarity or relation. Each group represents an equivalence class where elements share this relation. The collection of all these equivalence classes partitions the original set. For instance, if we have a set of students and an equivalence relation based on the same grade, the students can be grouped into classes based on their grade levels, thereby partitioning the entire student set.

Think of sorting a basket of mixed fruits by type. You can create subsets of apples, oranges, and bananas. Each subset not only represents a group of similar fruits but also partitions the basket based on the types of fruits it contains. This demonstrates how an equivalence relation, like 'is the same type of fruit', creates distinct subsets (or classes) that collectively make up the full basket.

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