Consider A=$\mathbb{Z}$. For each integer $n$, define

$$B_n = \{{m\in \mathbb{Z}\ | \ (\exists q)(m=n+5q)}\}.$$

Prove that $\{{B_n}\}_{n\in\mathbb{Z}}$ is a partition of $\mathbb{Z}$. Identify the equivalence classes.


closed as off-topic by Andrés E. Caicedo, José Carlos Santos, Lee Mosher, zipirovich, Jack Dec 9 '17 at 21:40

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Lee Mosher, zipirovich, Jack
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Are $B_n$ and $B_{n-5}$ disjoint . Because $m = n + 5q = n-5 + 5(q+1)$. So they are not disjoint. Unless in the definition something is missing. $\endgroup$ – Chirantan Chowdhury Dec 9 '17 at 17:57
  • 1
    $\begingroup$ Welcome to stackexchange. You are more likely to get answers rather than votes to close if you edit the question to show us what you have tried and where you are stuck. $\endgroup$ – Ethan Bolker Dec 9 '17 at 18:00
  • $\begingroup$ @ChirantanChowdhury Those sets are identical, which is not a problem. (If they weren't identical they would have to be disjoint and your objection would be relevant.) $\endgroup$ – Ethan Bolker Dec 9 '17 at 18:02
  • $\begingroup$ So we need to consider only $B_1,B_2,B_3,B_4,B_5$ which are disjoint by the division algorithm applied to a number divided by 5. $\endgroup$ – Chirantan Chowdhury Dec 9 '17 at 18:06
  • $\begingroup$ This is NOT a do-my-homework-for-me site. Within an hour you've posted two questions that are just copy-and-pasted texts of probably homework questions, without adding anything from yourself at all. If you're unwilling to make any effort on your homework, why should other people do that for you? $\endgroup$ – zipirovich Dec 9 '17 at 18:13

$B_n$ is the set of all the numbers that is congruent to $n$ modulo $5$. Now obviously

  1. Reflexive: $a\equiv a~(mod~5)$,
  2. Transitive: $a\equiv b~(mod~5)$ and $b\equiv c~(mod~5)$ implies $a\equiv c~(mod~5)$,
  3. Symmetric: $a\equiv b~(mod~5)$ implies $b\equiv a~(mod~5)$
  • $\begingroup$ These are correct statements. They don't directly answer the OP's questions. $\endgroup$ – Ethan Bolker Dec 9 '17 at 18:06

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