1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with equivalent elements?
2) What is the criteria for something to be an equivalence class? If the set of natural numbers was partitioned into odds {O} and evens {E}, then if I say E_1 ~ E_2 etc, am I just saying that they are equivalent in terms of how they were partitioned (i.e. divisible by 2)?
Does that mean I could partition the real numbers into subsets with equal numbers of decimal places? If that is a legitimate partition, how do I 'mathematically' formulate it? If it is not, why not?
3) If a set is partitioned into non-overlapping, non-empty subsets at random, how would there be equivalence on the partitioned subsets?
Best Answer
The idea for the first one is the following: Given a partition $\Delta$ of $X$, that partition induces an equivalence relation on $X$. Namely, we say $x$ and $y$ stand in relation if they belong to the same set of the partition. You can check that this is indeed an equivalence relation. Because
$(1)$ $x$ is always in the same set as $x$.
$(2)$ If $x$ is in the same set as $y$, then $y$ is in the same set as $x$.
$(3)$ If $x$ is in the same set as $y$, and $y$ is in the asme set as $z$, then $x$ is in the asme set as $z$.
Given a relation $\sim $ on a set $X$, we say it is an equivalence relation if it has the following three properties:
$(1)$ Reflexivity $x\sim x$. That is, every element stands in relation with itself.
$(2)$ Symmetry If $x\sim y$, then $y\sim x$. That is, if $x$ stands in relation with $y$, then $y$ stands in relation to $x$.
$(3)$ Transitivity If $x\sim y$ and $y\sim z$, then $x\sim z$. That is if $x$ stands in relation with $y$ and $y$ stands in relation with $z$, then $x$ stands in relation with $z$.
You can check that the relation divisibility is transitive and reflexive. You an check the relation of inclusio is transitive and reflexive, too, but not symmetryc. You can check that congruence $\mod{}$ a number $n$ is a equivalence relation, and so is usual equality of numbers.
Given a relation $\sim$ on a set $X$, and an element of $X$, we define the equivalence class $[[x]]$ of $x$ as all elements in $X$ that stand in relation to $x$. That is
$$[[x]]=\{a\in X:a\sim x\}$$
You can check equivalence classes are:
$(1)$ Non-empty: For each $x$, $x\in [[x]]$.
$(2)$ Either disjoint or equal: If there is one element $p$ in $[[x]]\cap[[y]]$ then we have that $p\sim y$ and $p\sim x$. Reflexivity means $y\sim p$ and $p\sim x$ so transitivity means $y\sim x$. Reflexivity once more gives $y\sim x$. Thus whenever $a\in [[x]]$, $a\sim x$ and $x\sim y$, whence $a\sim y$. Similarily whenever $b\in [[y]]$, $b\sim y$ and $y\sim x$, whence $b\sim x$. This means $[[x]]=[[y]]$.
This means that they are a partition of $X$. So any partition induces an equivalence relation, and conversely.
As to the first thing you say: if by unique you mean $x\sim y\iff x=y$, then what you have is the equivalence relation of equality which simply induces the partition of $X$ into singletons of its elements.
On you last question you talk about an equivalence on the partitioned subsets. I presume you mean an equivalence relation , but what do you mean by "on the partitioned subsets"?
Regarding the part on the "partitioning the real numbers into subsets of equal number of decimal places." Recall there are some rational numbers with nonterminating decimal expansion, and any irrational number will have a nonterminating decimal expansion. Thus, maybe you will have to work with rationals whose decimal expansion has the same amount of decimal numbers, and say talk about "number of decimal places" with nonterminating rationals acording to the period of the expansion. For example, $3^{-1}=0.\hat 3$. Else, if the number is irrational, it goesto $\mathbb I$, the set of irrational numbers.