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Mathematics LibreTexts

2.2 Equivalence Relations and Partial order

Definition

An equivalence relation is a relation that is reflexive(R), symmetric(S) and transitive(T).

Example \(\PageIndex{1}\): \(=\)

 

Definition

Given an equivalence relation \( R \) over a set \( S, \) for any \(a \in S \) the equivalence class of a is the set \( [a]_R =\{ b \in S  \mid a R b \} \), that is 
\([a]_R \) is the set of all elements of S that are related to \(a\).

Example \(\PageIndex{2}\):

Define a relation that two shapes are related iff they are the same color.  Is this relation an equivalence relation?

Equivalence classes are:

Example \(\PageIndex{3}\): 

Define a relation that two shapes are related iff they are similar.  Is this relation an equivalence relation?

Equivalence classes are:

Theorem:

If \( R \)  is an equivalence relation over \(S\), then every \( a \in S\)  belongs to exactly one equivalence class .

Definition

A binary relation is a partial order if and only if the relation is reflexive(R), antisymmetric(A) and transitive(T).

Example \(\PageIndex{4}\): \( \leq\)

 

 

Definition

Hasse diagram