Math and science::Algebra::Aluffi

# Equivalence relations

### The datum of an equivalence relation is a [...]

What is the information that is contained by an equivalence relation on a set?

The datum of an equivalence relation on a set $$S$$ turns out to be [something of $$S$$]. Compare to the case of the product of two sets, which has a set of ordered pairs as its datum.

### [...]. Definition.

Let $$S$$ be a set. A [...] of $$S$$ is a family of [something something] subsets of $$S$$ that [...].

For example, $$\{\{1,3,5\}, \{2,4,6\}, \{0\}, \{7\}, \{8, 9\} \}$$ is a [...] of the set $$\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$.

### From equivalence relations to partitions, and back

There is a 1-1 correspondence between partitions and equivalence relations, and in this sense they represent the same notion.

Here is the process of obtaining a partition from an equivalence relation. Let $$\sim$$ be an equivalence relation on a set $$S$$. For every $$a \in S$$ the equivalence class of $$a$$ is the subset of $$S$$ defined by

[$[a]_{\sim} = \quad ?$]

The set of all equivalence classes is a partition of $$S$$, denoted $$\mathcal{P}_{\sim}$$.

Conversely, given a partition $$\mathcal{P}$$, we can form an equivalence relation $$\sim$$ where $$a \sim b$$ is true if $$a$$ and $$b$$ are in the same element of $$\mathcal{P}$$.