Given the equivalence relation $x \sim y \iff x - y$ is an even integer, we want to determine the structure of the equivalence classes under this relation.

Step-by-step breakdown:

1. Set of Rational Numbers: Here, $X = \mathbb{Q}$, meaning that $x$ and $y$ are rational numbers. So $x, y \in \mathbb{Q}$.

2. Equivalence Condition: The condition $x \sim y \iff x - y$ is an even integer. We interpret this as follows:

   • For $x \sim y$, the difference $x - y$ must be an integer, specifically an even integer. This restricts the relation to certain rational numbers.

3. Form of the Rational Numbers: Let’s express $x$ and $y$ in their general rational form: $x = \frac{a}{b}, \quad y = \frac{c}{d} \quad \text{where} \ a, b, c, d \in \mathbb{Z} \quad \text{and} \ b, d \neq 0.$ For $x - y$ to be an even integer, we need: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ For this to be an even integer, the numerator $ad - bc$ must be an even integer, and the denominator $bd$ must divide this even integer cleanly, yielding an integer.

4. Conclusion - Equivalence Class: The equivalence class of any rational number $x = \frac{a}{b} \in \mathbb{Q}$ under this relation consists of all rational numbers $y = \frac{c}{d} \in \mathbb{Q}$ such that: $x - y \ \text{is an even integer}.$ In particular, each equivalence class can be described by the congruence of $x$ and $y$ modulo 2 when expressed as fractions. This means two rational numbers $x$ and $y$ are in the same equivalence class if their difference is even.

The equivalence class of a rational number $x \in \mathbb{Q}$ is the set of all rational numbers that differ from $x$ by an even integer.

Example:

For $x = 1/2$, the equivalence class would be all rational numbers $y = \frac{c}{d} \in \mathbb{Q}$ such that $\frac{1}{2} - y$ is an even integer. Thus, one such equivalence class might contain numbers like $\frac{1}{2}, \frac{3}{2}, -\frac{1}{2}$, etc., since the difference between these numbers and $1/2$ is an even integer.

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Would you like more details on specific equivalence classes or examples?

Related Questions:

1. How do equivalence relations partition a set?
2. Can you explain the difference between equivalence classes of rationals vs integers?
3. How do we find representatives for each equivalence class in general?
4. What would the equivalence class of $\frac{3}{4}$ be under this relation?
5. Can you provide an example of a non-equivalence relation with similar conditions?

Tip: Equivalence relations divide a set into mutually exclusive equivalence classes, where every element in the set belongs to exactly one class.