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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?

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up vote 2 down vote accepted

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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So for the element 0, the equivalent class would be the subset {(0,0)}? – user1766888 Nov 10 '12 at 18:47
@user1766888: No, the equivalence classes are subsets of the underlying set $\{0,1,2,3\}$, not of the relation. The equivalence class of $0$ is $\{0\}$, since $0$ is the only thing related to $0$ by that relation. – Brian M. Scott Nov 10 '12 at 18:50
Brian 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. – DonAntonio Nov 10 '12 at 18:52

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, $(a,x)\in R$.

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