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 September 22nd, 2018, 10:01 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, 10:07 AM #2 Senior Member   Joined: Oct 2009 Posts: 863 Thanks: 328 The equivalence relation is defined on $\mathbb{Z}$ and is defined as $x\sim y$ if and only if $x-y$ is divisible by $4$. Can you for example find out what elements are equivalent to $0$ now? Thanks from Roberto 37 September 22nd, 2018, 10:15 AM   #3
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Quote:
 Originally Posted by Roberto 37 hi, good afternoon how to construct the equivalence class of z4 ? i know it is 0,1,2,3 but because?
Not quite, those are just representatives of the equivalence classes.

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 $, , $ and $$ partition $\mathbb{Z}$. That is, prove that every integer is in exactly one of those four sets. Tags classes, equivalence Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post LaC0saNostra Applied Math 4 October 27th, 2012 04:02 PM guynamedluis Real Analysis 0 November 21st, 2011 08:39 PM liangteh Applied Math 8 March 12th, 2011 04:31 PM Zhai Applied Math 9 May 5th, 2010 01:18 PM

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