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September 22nd, 2018, 11:01 AM   #1
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Equivalence classes

hi, good afternoon

how to construct the equivalence class of z4 ?

i know it is 0,1,2,3 but because?
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September 22nd, 2018, 11:07 AM   #2
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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?
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September 22nd, 2018, 11:15 AM   #3
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Quote:
Originally Posted by Roberto 37 View Post
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.
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