Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

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.

share|improve this answer
1  
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, $(a,x)\in R$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.