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equivalence_relation [2007/03/30 20:36] vkuncak |
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| ====== Equivalence relation ====== | ====== Equivalence relation ====== | ||
| - | An equivalence relation $\sim$ is a binary relation on set $A$ (that is, a subset of $A^2$) that is reflexive, symmatric, and transitive, that is, the following three properties hold: | + | An equivalence relation $\sim$ is a binary relation on set $A$ (that is, a subset of $A^2$) that is reflexive, symmetric, and transitive, that is, the following three properties hold: |
| * $x \sim x$ | * $x \sim x$ | ||
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| * $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$ | * $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$ | ||
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| + | Given an equivalence relation $\sim$, we define the set of equivalence classes $A_{\sim}$ by | ||
| + | \begin{equation*} | ||
| + | \end{equation*} | ||
| + | |||
| + | The equivalence classes of a non-empty set form a [[partition]] of $A$. | ||
| + | |||
| + | Conversely, given a [[partition]] $P \subseteq 2^A$ of set $A$, the relation defined by | ||
| + | \begin{equation*} | ||
| + | x \sim y\ \iff\ \exists S \in P. \{x,y\} \subseteq S | ||
| + | \end{equation*} | ||
| + | is an equivalence relation such that $A_{\sim} = P$. | ||