The above relation is equivalent for the set {0,1,2,3}. How would you find the equivalence class for this relation or any general relational set of pairs of integers?
Equivalence Class of the relation {(0,0) (1,1) (2,2) (3,3)}
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2 Answers
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Check that each element of $\,\{0, 1,2,3\}\,$ belongs to exactly one singleton subset of the given relation and, thus, this equiv. relation is equivalent (isomorphic, if you will) with the equality relation.
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0Brian is correct and I miswrote (or, if you will, wrote something that can easily be misinterpreted): every element of the given set is related only to itself and thus only the singleton of the *given set* (not of the relation!) containing it is its equiv. class. – 2012-11-10
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Let the relation be called $R$. Then, $(a,b)\in R$ means that the pair $(a,b)$ is in relation $R$, and this fact is also often denoted simply by $aRb$. Now this $R$ relation is just the equality relation on the base set $\{0,1,2,3\}$.
In general, if $R$ is an equivalence relation, an element $x$ belongs to the equivalence class of another element $a$ iff $x$ and $a$ are 'equivalent according to $R$', that is, $aRx$, in other words, $(a,x)\in R$.