What does "relation induced by a partition" mean?

Question

The question is: What is the equivalence relation R, induced by the partition P of A? (A, P are given)

I don't get what "induced by" means.

Answer 1

There is a particular natural connection between equivalence relations and partitions. Each partition corresponds to an equivalence relation, and each equivalence relation corresponds to a partition. Going back and forth along this correspondence will get you back where you started.

The correspondence is this: Given a partition, each element is related to each element in the same part, and nothing else. The other way: Given an equivalence relation, a part of the partition is given by a maximal set of elements all related to one another.

This correspondence is what they mean by "the equivalence relation induced by the partition".

I suspect that they use the word "induce" because the correspondence is based on a concrete construction to get from one to the other. Not all such natural correspondences are like that.

Answer 2

The partition $P$ is a set $\{A_\lambda\mid\lambda\in\Lambda\}$ of subsets of $A$. The relation $R$ is the one defined by$$a\mathrel Ra'\iff\text{for some }\lambda\in\Lambda,\ a,a'\in A_j.$$

Answer 3

That would mean that you think of the partition $P$ as giving the equivalence classes of the relation $R$, so $aRb$ if and only if there is an $S \in P$ such that $a, b \in S$.

In this way, equivalence relations and partitions are kind of interchangeable - any equivalence relation corresponds exactly to a partition of the set, which is the equivalence classes.

Answer 4

Let $P$ be a partitioning of $X$. Then it induces an equivalence relation $\sim_P$ such that $x\sim_P y$ if and only if $x,y$ belong to the same element of $P$.

Conversly if $\sim$ is an equivalence relation then it induces a partitioning $P_\sim:=\{[x]_\sim\ |\ x\in X\}$ where $[x]_\sim=\{y\in X\ |\ y\sim x\}$.

The nice property of these operations is that they are inverses of each other:

$$P_{\sim_P}=P$$ $$\sim_{P_\sim}=\sim$$

In other words partitionings and equivalence relations are the same thing.