What is the Quotient Set for the Given Equivalence Relation?

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Discussion Overview

The discussion revolves around the concept of quotient sets in the context of equivalence relations defined on the integers. Participants explore specific equivalence relations, their resulting equivalence classes, and the implications of including negative integers. The conversation also touches on a second problem involving a different equivalence relation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes an equivalence relation on integers defined by 3a + b being a multiple of 4, detailing the equivalence classes for 0, 1, 2, and 3.
  • Another participant agrees with the initial claim and notes that the equivalence classes correspond to classes "mod 4," mentioning the relationship between 3 and 4 being relatively prime.
  • A participant questions whether negative integers need to be included in the equivalence classes.
  • The same participant later reflects that they have already accounted for negative integers in their previous statements.
  • Another participant introduces a new problem involving the equivalence relation defined by a^2 - b^2, suggesting a partition into multiples of 3 and non-multiples of 3.
  • A participant seeks clarification on the definition of the equivalence relation in the context of the new problem, expressing confusion about the relevance of multiples of 3.
  • One participant confirms that -3 is indeed a multiple of 3 and clarifies the definition of multiples in relation to negative numbers.

Areas of Agreement / Disagreement

There is some agreement on the equivalence classes defined by the first relation, but the second problem introduces confusion and differing interpretations regarding the equivalence relation and its implications. The discussion remains unresolved regarding the second problem.

Contextual Notes

Participants express uncertainty about the definitions and implications of the equivalence relations, particularly in the second problem. There are also unresolved questions about the treatment of negative integers in the context of equivalence classes.

Caldus
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If I have an equivalence relation acting on all integers (Z): a ~ b if any only if 3a + b is a multiple of 4, then here is what I think the quotient set is:

The equivalence class of 0 = {x belongs to Z | x ~ 0} = {x | 3x = 4n for some integer n}. (The set would look like {0, 4, 8, 12, 16...}.)

The equivalence class of 1 = {x belongs to Z | x ~ 1} = {x | 3x + 1 = 4n for some integer n}. (The set would look like {1, 5, 9, 13, 17...}.)

The equivalence class of 2 = {x belongs to Z | x ~ 2} = {x | 3x + 2 = 4n for some integer n}. (The set would look like {2, 6, 10, 14, 18...}.)

The equivalence class of 3 = {x belongs to Z | x ~ 3} = {x | 3x + 3 = 4n for some integer n}. (The set would look like {3, 7, 11, 15, 19...}.)

So based on that, I conclude that there are 4 elements in the quotient set. Each element contains one of the sets above.

Am I accurate here? Thanks.
 
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Yes, you are correct. And did you notice that the equivalence classes are precisely the equivalence classes "mod 4"? Since 3 and 4 are relatively prime, If 3x is divisible by 4, then x is divisible by 4: if 3a and b are congruent mod 4, then so are a and b.
 
I just realized something. Don't I need to include negative numbers as well?
 
Nevermind, I already took care of them didn't I?
 
Also, for another problem: a^2 - b^2 acting on Z (a ~ b)...

Is the partition for this problem going to be split into two parts:
1. Numbers that are multiples of 3 (0, 3, 6, 9, 12...)
2. Numbers that are not multiples of 3 (1, 2, 4, 5, 7, 8...)

(Also, is -3 considered a multiple of 3? I'm getting myself confused here...lol...)

Am I accurate again here?

Thank you for your help.
 
What, exactly, is the problem?
"a^2 - b^2 acting on Z (a ~ b)..." are you saying that a~b if and only if a^2- b^2= 0? Or are a and b the equivalence classes defined before?

In either case I don't see what being a multiple of 3 has to do with anything.

(And, yes, -3 is a multiple of 3. When you are including negative numbers, a is a multiple of b if and only if a= nb for some integer n.)
 

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