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.

R is a relation defined on the integers by $(a,b) \in R$ is $a^2-b^2$ and is divisible by 3.

I set a or b to zero to get all the negative and positive values in the equivalence class. Although I want to say it is $(...,-9n,-6n,3n,0,3n,6n,9n,...)$ for some integer n. But I do not think this is correct because if n = 1, 6n does not belong to the relation. What am I doing wrong?

share|improve this question
    
If $a=3n,b=3m$ and if $a=3n\pm1, b=3m\pm1$ –  lab bhattacharjee Nov 21 '13 at 4:27

1 Answer 1

We have your equivalence relation R as $a \sim b$ if $3|(a^2 - b^2)$.

To find the class of elements equivalent to $0$, we need to set one of the elements to $0$ (just one because of reflexivity, symmetry, and transitivity, as this is an equivalence relation): suppose $a \sim 0$, i.e. $3|(a^2 - 0^2) \Longrightarrow 3|a^2$. Then by Euclid's Lemma, $3|a$. So the equivalence class of $0$ is the set of all integers that we can divide by $3$, i.e. that are multiples of $3: \{\ldots, -6,-3,0,3,6, \ldots\}$.

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.