September 22nd, 2018, 11:01 AM  #1 
Newbie Joined: Jun 2018 From: Brasil Posts: 20 Thanks: 0  Equivalence classes
hi, good afternoon how to construct the equivalence class of z4 ? i know it is 0,1,2,3 but because? 
September 22nd, 2018, 11:07 AM  #2 
Senior Member Joined: Oct 2009 Posts: 628 Thanks: 190 
The equivalence relation is defined on $\mathbb{Z}$ and is defined as $x\sim y$ if and only if $xy$ is divisible by $4$. Can you for example find out what elements are equivalent to $0$ now? 
September 22nd, 2018, 11:15 AM  #3  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 284 Thanks: 86 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
Given an integer $x$, the equivalence class of $x$ is the set $[x]$ of all integers equivalent to $x$ (where here equivalent means congruent mod $4$). That is, $[x] = \{ n \in \mathbb{Z} \mid n \equiv x \bmod 4 \}$. But $n \equiv x \bmod 4$ if and only if $n = x + 4y$ for some integer $y$, so we see $[x] = \{x + 4y \mid y \in \mathbb{Z} \}$. See if you can now prove that $[0], [1], [2]$ and $[3]$ partition $\mathbb{Z}$. That is, prove that every integer is in exactly one of those four sets.  

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