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The "≡ mod 3" relation is an equivalence relation on the set {1,2,3,4,5,6,7}. List the equivalence classes.

I understand "≡ mod n" relation on Z is transitive. I just cant see how to start this problem, Thanks for any tips.

J Crawford
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    Two integers are congruent modulo $3$ if and only if they share the same remainder when divided by $3$. – Dave Jun 07 '17 at 01:40

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Elements which can be written as $3k+1$:

$$[1]=\{ 1,4,7 \}$$

Elements which can be written as $3k+2$: $$[2]=\{ 2, 5\}$$

Can you write down the set of elements which can be written as $3k$?

Siong Thye Goh
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  • 3k+1 [1]= {1,4,7} 3k+2 [2] = {2,7} 3k+3 [3]={3} Repeats with single digit from there. – J Crawford Jun 07 '17 at 02:03
  • $[3]={ 3,6}$, there are only $3$ equivalence class. – Siong Thye Goh Jun 07 '17 at 02:06
  • Oh opps thank you! So when I am looking at a question in the future like this it is basically the format when Mod 3, "3k+x" right? With x increasing by 1 and K part of the elements? – J Crawford Jun 07 '17 at 02:10
  • In general, I will pick an element, say $1$, find all element that is equivalent to it, that would form an equivalence class. Then I will move on to another element that is not in any equivalence class yet. Repeat. – Siong Thye Goh Jun 07 '17 at 02:11
  • Okay this helps a lot thank you, the book I have is really bad. – J Crawford Jun 07 '17 at 02:19