How do I describe a partition determined by $R$ by listing pieces?

Question

$R$ is the relation $\{(1,1),(2,3),(2,2),(3,2),(3,3)\}$ on the set $\{1, 2, 3\}$. It then is reflexive, symmetric, transitive and thus an equivalence relation. How do I describe the partition formed by $R$ by listing the pieces in the partition?

Answer 1

The "pieces of the partition" that arise from an equivalence relation $R$ given on a set are usually called the equivalence classes. To describe an equivalence class of an equivalence relation $R$, it is enough to specify any one member of the equivalence class.

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For your question, there are two equivalence classes, namely, $$ \{ 1 \} \text{ and } \{ 2, 3 \}. $$

An equivalence class is usually denoted using box brackets, $[\cdot]$. So, one can list the partition of $\{ 1,2,3 \}$ under the given equivalence relation $R$ as $$ [1] \text{ and } [2] $$ or $$ [1] \text{ and } [3]. $$ To elaborate, if $R$ is an equivalence relation on a set $S$, then for each $x \in S$, the notation $[x]$ is defined as follows: $$ [x] = \{ y \in S : xRy \}. $$

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Note that the equivalence classes will change as your equivalence relation changes. Based on the discussion in the comments, it appears that you were uncertain whether the partition $$ [1],[2],[3] $$ would work as an answer to the current question. It does not, because they describe a partition arising from a different equivalence relation on the set $\{ 1,2,3\}$, namely $$ xRy \iff x = y. $$