# Describe an Equivalence Relation Given A Partition (Rudin Analysis Problem 6.15)

Let $$S=\{a,b,c,d\}$$ and let $$\mathscr{P}=\{ \{ a \}, \{ b,c \}, \{ d \} \}$$. Note that $$\mathscr{P}$$ is a partition of $$S$$. Describe the equivlanece relation $$\textbf{R}$$ on $$S$$ determined by $$\mathscr{P}$$ as indicated in Theorem 6.15.

Theorem 6.15 states, "Let $$\textbf{R}$$ be an equivalence relation on a set $$S$$. Then $$\{ E_x:x \in S \}$$ is a partition of $$S$$. The relation 'belongs to the same piece as' is the same as $$\textbf{R}$$. Converseley, if $$\mathscr{P}$$ is a partition of $$S$$, let $$\textbf{R}$$ be defined by $$x\textbf{R}y$$ iff $$x$$ and $$y$$ are in the same piece of the partition. Then $$\textbf{R}$$ is an equivalence relation and the corresponding partition into equivalence classes is the same as $$\mathscr{P}$$"

A relation acts on a set if $$(x,y)\in \textbf{R}$$. Since $$\mathscr{P}$$ has three members that must mean there are three pieces of the partition (and thus three seperate equivalence classes). $$E_h=\{y \in S:y\textbf{R}h \}$$, $$E_i=\{y\in S: y\textbf{R}i \}$$, $$E_j=\{y\in S: y \textbf{R} j \}$$.

Each piece of the partition (i think) can be used to construct the ordered pairs of the members of the relation $$\textbf{R}$$ by simply using the members of each equivalence class in all possible combinations of ordered pairs similar to below.

$$\mathscr{P}=\{ \underbrace{\{ a \}}_{(a,a)}, \underbrace{\{ b,c \}}_{(b,b),(b,c),(c,b),(cc)}, \underbrace{\{ d \}}_{(d,d)} \}$$

This yields the relation to be $$\textbf{R}=\{(a,a),(b,b),(b,c),(c,b),(c,c),(d,d) \}$$ which according to the book is the correct answer. However I pretty much worked this out going backwards (starting from the answer and working my way to the original practice problem) so I am unsure if my understanding of the difference between pieces of a partition and thee members of the equivalence class are correct. Is there a more formal way to figure out the equivalence relation $$\textbf{R}$$?

• what is $\{E_z:x\in S\}$ supposed to mean? – Pink Panther Jul 30 '19 at 17:44
• It seems your understanding is correct. For each piece of the partition, make all possible pairs of their members; the equivalence class is the union of these sets of pairs. It is actually very simple. – amrsa Jul 30 '19 at 17:46
• Typo my apologies, it actually says. "Then $\{E_x:x \in S \}$ is a partition of $S$." – kristhedemented Jul 30 '19 at 17:46

The equivalence relation $$\mathbf{R}$$ is defined by

for all $$x,y\in S$$, $$(x,y)\in\mathbf{R}$$ if and only if there exists $$E\in\mathscr{P}$$ such that $$x,y\in E$$.

This is easily seen to be an equivalence relation that induces the same partition on $$S$$.

Thus $$\mathbf{R}=\bigcup_{E\in\mathscr{P}}(E\times E)$$