Chenyang Yuan

Book Review: Naive Set Theory by Paul Halmos (Part 1)

Last weekend I just finished reading Naive Set Theory by Paul Halmos, and I think it would be good to write down some thoughts and interesting proofs in the book while the material is still fresh in my mind.

Overall I think this book is a short and concise introduction to set theory, covering a lot of material in only 102 pages. There is a good summary and review of the book here, so I’m not going to repeat what that author said. For me, one major shortcoming of this book is that there are too few exercises, averaging only 2-3 per chapter. I think best (and only?) way to learn mathematics is to come up with examples and work out exercises, both of which this book lack. I guess the Halmos expected readers to come up with such examples themselves:

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

As a result, I took a very long time to read each chapter, only looking at the bare minimum of definitions and propositions and trying to come up with proofs myself before looking at the book’s. While working through the book, I’ve also summarized the definitions and proofs written them down in my notebook, which I will (eventually) scan and upload digitally. So instead of providing a summary (the above link already did a good job doing that), I’m going to pick my most favorite proofs and explain them in the following posts.

Counting Equivalence Classes

A relation $R$ is a set of ordered pairs. For every $(x, y) \in R$, we can write $x R y$. A relation in set $X$ is a subset of $X \times X$, a relation between members of $X$. An equivalence relation is a relation in set $X$ where the following three properties hold:

1. Reflexive: $x R x$
2. Symmetric: $x R y \iff y R x$
3. Transitive: $x R y \wedge y R z \Rightarrow x R z$

For all $x, y, z \in X$. We can represent equivalence relations in a finite set by using a table listing all possible ordered pairs, representing an equivalence with an entry in the table.

  a b c d      a b c d      a b c d      a b c d
a o o        a o          a o o o      a o   o
b o o        b o o        b o o        b   o
c     o      c o o o      c o   o      c o   o
d            d       o    d       o    d       o


The first three relations are each missing one property of equivalence relations: The first relation is not reflexive ($d \cancel{R} d$), the second one is not symmetric ($b R a$ but $a \cancel{R} b$), and the third one is not transitive ($c R a$ and $a R b$ but $c \cancel{R} b$). The last relation is satisfies all three properties and is an equivalence relation.

Given a set, we want to count the total number of equivalence relations in it. Let’s try to list out the number of equivalence relations in a set of 4 elements. In the figure below, we represent each equivalence relation in a grid, with each colored square indicating an ordered pair. The reflexive pairs in each relation ($aRa$, etc) are greyed for clarity.

We can immediately see some interesting patterns from the figure. An equivalence relation is like splitting the elements into groups, with elements in each group mutually equivalent. The top row represents splitting each element into a group on its own $(\{a\}, \{b\}, \{c\}, \{d\})$, or into one group with 4 elements $(\{a, b, c, d\})$. The second row represents splitting into two groups of two elements each $(\{a, b\}, \{c, d\}$ or $\{a, c\}, \{b, d\}$ or $\{a, d\}, \{b, c\})$, and so on. We call these groups equivalence classes; an equivalence class of $x \in X$ is $\{y \in X : x R y \}$.

From this, it’s natural to postulate that the number of equivalence relations is the same as the number of partitions in a set. A partition of set $X$ is a set of non-intersecting subsets of $X$ whose union is $X$. We first prove that if $R$ is an equivalence relation in $X$, the set of equivalence classes is a partition of $X$ such that $x R y$ iff they belong to the same subset in the partition.

Let’s call the set of equivalence classes $C$. If $x R y$, then they both belong to the same equivalence class in $C$. We have to prove that this is a partition. Since an equivalence relation is reflexive, $x R x$, so every $x \in X$ is in a equivalence class, and therefore $\cup C = X$. Now we prove that the sets in $C$ are non-intersecting. Suppose that $x$ belonged to two distinct equivalence classes $A$ and $B$. However, $\forall a \in A \land b \in B , a R b$, since $x R a \land x R b \Rightarrow$ $a R x \land x R b \Rightarrow$ $a R b$. Thus $A$ and $B$ are the same equivalence class, and this leads to a contradiction. Thus each $x \in X$ can only belong to one equivalence class, the proof is complete.

The next step is to prove that for every partition $C$ of $X$, if we can define a relation such that $x R y$ iff they belong to the same subset in the partition, the set of equivalence classes of $R$ is exactly $C$. This proof is left as an exercise to the reader. I find this connection between equivalence classes and partitions interesting as it can be easily visualized from the diagram above. As to how to count the number of equivalence classes/partitions, this wikipedia article gives a good overview.

That’s it for this post! Stay tuned for the next one, where I’m going to write about well-ordering and transfinite recursion.