FOM Book: 6.1 (Partitions & Equivalence Relations)

We’ve been talking about this topic for a while in FOM, so I guess it’s about time to read about it in the book and put some formal definitions out there.

Definition 6.12: For a set, A, a partition of A is a set C of subsets of A (i.e. C ⊆ P(A)) such that:

a) (∀x∈A)(∃U∈C)(x∈C)

b) (∀U∈C)(U ≠ ∅)

c) (∀U,V ∈ C)(U∩V≠∅ ⇒ U = V)

In real life words, this basically means that, based on the a) property, every element in A falls into some set C that A is split into. (Think of one of those little kid plates that have the different sections for foods. A = the whole plate, and every element of food of going to end up in some section of the plate, the sections being the C subsets)

According to the b) property, no partitions of a set are the empty set. (It wouldn’t make sense for a kid’s plate to have an food slot that doesn’t technically exist..). This property just works to avoid any unneccesary complications using an empty set, becuase it adds nothing of important to a partition set.

And the last property just talks about the relationship between U and V as partitions. Using the kid’s plate example again, the c) property states that if you’re looking at two food slots and they have some type of food in common, you’re really only thinking of one food slot. 

When combining a) and c), we can say that each element of A belongs to exactly one of the parts of the partition. It’s also helpful to mention that: U,V⊆A. It’s intuitive in the kid’s plate example, because the slots are what make up the plate.

Definition 6.14: For a set A, an equivalence relation on A is a relation on A which is reflexive, symmetric, and transitive.

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