# Are These Equivalence Classes Correct for the Given Relation?

• Dustinsfl
In summary, a relation problem is a type of problem that involves understanding and describing relationships between variables or individuals/groups. Examples include determining correlation, analyzing dynamics, and resolving conflicts. A relation is a connection or association between things, often represented in mathematics as ordered pairs. Defining a relation problem is important for identifying the issue and finding a solution, and strategies for solving it may include identifying variables, analyzing the relationship, and seeking outside help or mediation.
Dustinsfl
Define the relation ∼ on ℤ as follows: For a,b ∈ ℤ, a∼b iff. 2a + 3b ≡ 0 (mod 5). The relation ∼ is an equivalence relation on ℤ. Determine all the distinct equivalence classes for this equivalence relation.
Reflexive if a∼a.
2a + 3a ⇒ 5a ≡ 0 (mod 5); therefore, the relation is reflexive.
Symmetric if a∼b, then b∼a.
2a + 3b ≡ 4(2a + 3b) ≡ 8a + 12b ≡ 3a + 2b ≡ 0 (mod 5); therefore, the relation is symmetric.
Transitive if a∼b and b∼c, then a∼c.
a∼b ⇒ 2a + 3b ≡ 0 (mod 5)
b∼c ⇒ 2b + 3c ≡ 0 (mod 5) By adding the two, we obtain ⇒ 2a + 5b + 3c ≡ 2a + 3c ≡ 0 (mod 5); therefore, the relation is transitive.
2a + 3b ≡ 0 (mod 5) ⇒ 5 | (2a + 3b) ⇒ 5m = 2a + 3b
[0] = {a ∈ ℤ | a∼0} = {a ∈ ℤ | 5m = 2a} = {a ∈ ℤ | 2a = 5m} = {..., 5, 10, 15, ...}
[1] = {a ∈ ℤ | a∼1} = {a ∈ ℤ | 5n = 2a + 3} = {a ∈ ℤ | 2a = 5n - 3} = {..., 1, 6, 11, ...}
[2] = {a ∈ ℤ | a∼2} = {a ∈ ℤ | 5p = 2a + 6} = {a ∈ ℤ | 2a = 5p - 6} = {..., 2, 7, 12, ...}
[3] = {a ∈ ℤ | a∼3} = {a ∈ ℤ | 5r = 2a + 9} = {a ∈ ℤ | 2a = 5r - 9} = {..., -2, 3, 8, ...}
[4] = {a ∈ ℤ | a∼4} = {a ∈ ℤ | 5t = 2a + 12} = {a ∈ ℤ | 2a = 5t - 12} = {..., -1, 4, 9, ...}

Are these correct?

Last edited:

Yes, I think they are correct. Nice job.

## 1. What is a relation problem?

A relation problem is a type of problem that involves understanding and describing the relationship between two or more variables. It can also refer to issues or conflicts between individuals or groups.

## 2. What are some examples of relation problems?

Examples of relation problems include determining the correlation between two sets of data, analyzing the dynamics of a romantic relationship, or resolving conflicts between coworkers.

## 3. How do you define a relation?

A relation is a connection or association between two or more things. In mathematics, a relation is often represented as a set of ordered pairs, where the first element in each pair is related to the second element.

## 4. What is the importance of defining a relation problem?

Defining a relation problem is important because it helps to clearly identify the issue at hand and provides a framework for finding a solution. It also allows for effective communication and collaboration when working towards a resolution.

## 5. What are some strategies for solving a relation problem?

Some strategies for solving a relation problem include identifying the variables involved, analyzing the nature of the relationship, and considering different perspectives. Other helpful approaches may include communication, compromise, and seeking outside help or mediation.

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