My Math Forum Equivalence classes

 Linear Algebra Linear Algebra Math Forum

 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: 544 Thanks: 174 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
Senior Member

Joined: Aug 2017
From: United Kingdom

Posts: 264
Thanks: 79

Math Focus: Algebraic Number Theory, Arithmetic Geometry
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 $[0], [1], [2]$ and $[3]$ partition $\mathbb{Z}$. That is, prove that every integer is in exactly one of those four sets.

 Tags classes, equivalence

 Thread Tools Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top