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I am trying to write a proof of the following-

Prove that $[a]_n=[b]_n$ if and only if $a \equiv b\pmod{n}$

I am not sure how to start or how to get to the final product. Any help will be appreciated.

stressed out
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Kerry Brennan
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  • How you should write down the proof pretty much depends on how those notations have been defined for you. It's generally a good idea to write down the definitions in your post when you ask a question. Not only it will show others that you're genuinely looking for help, but it also helps you get answers that are more suited for your class when your paper is graded. – stressed out Jan 27 '19 at 18:32
  • That's a general property of equivalence relations: equivalence classes make up a partition of the set on which the equivalence is defined. – Bernard Jan 27 '19 at 18:33

1 Answers1

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Welcome to MSE! Maybe you mean that the classes of $a$ and $b$ are equal iff $a\equiv b\mod n$. The class of $a$ is $\bar a = \{b\mid a\equiv b\mod n\}$.

First, let $a\equiv b\mod n$. One needs to show that $\bar a= \bar b$. For this, let $c\in\bar a$. Then $c\equiv a\mod n$. By transitivity, $c\equiv b\mod n$ and so $c\in\bar b$. Thus $\bar a\subseteq \bar b$. Similarly, $\bar b\subseteq \bar a$. Hence, $\bar a=\bar b$.

Conversely, if $\bar a= \bar b$, then $b\in\bar b=\bar a$ and so $a\equiv b\mod n$. Done.

Wuestenfux
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