Michael Hong wrote:

> Puzzle 32

> The definition of a partition in _Seven Sketches_ says:

> > **Definition.** If \\(A\\) is a set, a partition of \\(A\\) consists of a set \\(P\\) and, for each \\(p \in P\\), a nonempty subset \\(A_p \subseteq A\\), such that

$$A=\bigcup_{p\in P}A_p \qquad\text{and}\qquad \text{if }p\neq q\text{ then }A_p\cap A_q=\emptyset $$

> > We refer to \\(P\\) as the set of **part labels** and if \\(p\in P\\) is a part label, we refer to \\(A_p\\) as the **\\(p\\)th part**. The equation above says that each element \\(a\in A\\) is in exactly one part.

This is indeed the key to Puzzle 32. I've added the full definition so we can all stare at it.

There's something "wrong" with this definition. More precisely, it's an okay definition, but it does _not_ yield a one-to-one correspondence between partitions and equivalence relations. Can someone see the problem?

> Maybe this wording is a bit off? This definition is saying a partition is the labeling set \\(P\\) with the subset of \\( A\\).

No, it's not saying that. It's saying a partition is the set of **part labels** \\(P\\) and, for each part label \\(p \in P\\), a subset \\(A_p \subseteq A\\) called a **part**, such that each element of \\(A\\) is in exactly one part.

> Puzzle 32

> The definition of a partition in _Seven Sketches_ says:

> > **Definition.** If \\(A\\) is a set, a partition of \\(A\\) consists of a set \\(P\\) and, for each \\(p \in P\\), a nonempty subset \\(A_p \subseteq A\\), such that

$$A=\bigcup_{p\in P}A_p \qquad\text{and}\qquad \text{if }p\neq q\text{ then }A_p\cap A_q=\emptyset $$

> > We refer to \\(P\\) as the set of **part labels** and if \\(p\in P\\) is a part label, we refer to \\(A_p\\) as the **\\(p\\)th part**. The equation above says that each element \\(a\in A\\) is in exactly one part.

This is indeed the key to Puzzle 32. I've added the full definition so we can all stare at it.

There's something "wrong" with this definition. More precisely, it's an okay definition, but it does _not_ yield a one-to-one correspondence between partitions and equivalence relations. Can someone see the problem?

> Maybe this wording is a bit off? This definition is saying a partition is the labeling set \\(P\\) with the subset of \\( A\\).

No, it's not saying that. It's saying a partition is the set of **part labels** \\(P\\) and, for each part label \\(p \in P\\), a subset \\(A_p \subseteq A\\) called a **part**, such that each element of \\(A\\) is in exactly one part.