Define a relation ∼ on R × R as
(x1, x2) ∼ (y1, y2) if x1x2 = y1y2.
(i) Check that ∼ is an equivalence relation.
(ii) Identify and draw the equivalence classes.

Let's break this problem down into two parts: 

### Part (i): Checking that ∼ is an equivalence relation.

We need to check if the given relation, defined by \((x_1, x_2) \sim (y_1, y_2)\) if \(x_1 x_2 = y_1 y_2\), satisfies the three necessary properties for being an **equivalence relation**: reflexivity, symmetry, and transitivity.

1. **Reflexivity**: A relation is reflexive if \((x_1, x_2) \sim (x_1, x_2)\) for all \((x_1, x_2) \in \mathbb{R} \times \mathbb{R}\).
   - For reflexivity, we need \(x_1 x_2 = x_1 x_2\), which is clearly true. Hence, the relation is reflexive.

2. **Symmetry**: A relation is symmetric if whenever \((x_1, x_2) \sim (y_1, y_2)\), we also have \((y_1, y_2) \sim (x_1, x_2)\).
   - Given that \((x_1, x_2) \sim (y_1, y_2)\) means \(x_1 x_2 = y_1 y_2\), we also have \(y_1 y_2 = x_1 x_2\), which shows that \((y_1, y_2) \sim (x_1, x_2)\). Hence, the relation is symmetric.

3. **Transitivity**: A relation is transitive if whenever \((x_1, x_2) \sim (y_1, y_2)\) and \((y_1, y_2) \sim (z_1, z_2)\), we have \((x_1, x_2) \sim (z_1, z_2)\).
   - If \((x_1, x_2) \sim (y_1, y_2)\), then \(x_1 x_2 = y_1 y_2\), and if \((y_1, y_2) \sim (z_1, z_2)\), then \(y_1 y_2 = z_1 z_2\). Therefore, we have \(x_1 x_2 = z_1 z_2\), which shows that \((x_1, x_2) \sim (z_1, z_2)\). Hence, the relation is transitive.

Since the relation satisfies reflexivity, symmetry, and transitivity, we conclude that ∼ is an **equivalence relation**.

---

### Part (ii): Identifying and Drawing the Equivalence Classes.

To identify the equivalence classes, we need to describe the set of all pairs \((x_1, x_2)\) that are related to a particular pair \((y_1, y_2)\) under the given relation.

Recall that \((x_1, x_2) \sim (y_1, y_2)\) means \(x_1 x_2 = y_1 y_2\). This means that all pairs \((x_1, x_2)\) in the equivalence class of \((y_1, y_2)\) must satisfy \(x_1 x_2 = y_1 y_2\). In other words, the equivalence class of \((y_1, y_2)\), denoted as \([ (y_1, y_2) ]\), consists of all pairs \((x_1, x_2)\) such that:

\[
x_1 x_2 = y_1 y_2
\]

This represents a hyperbola in the plane when \(y_1 y_2 \neq 0\). Each equivalence class is essentially the set of all points \((x_1, x_2)\) whose product is equal to some constant \(c = y_1 y_2\).

- If \(y_1 y_2 = c\), then the equivalence class is the set of all pairs \((x_1, x_2)\) such that \(x_1 x_2 = c\). Geometrically, this is a **hyperbola** in \(\mathbb{R}^2\) defined by the equation \(x_1 x_2 = c\).
  
- If \(y_1 y_2 = 0\), then either \(y_1 = 0\) or \(y_2 = 0\). The equivalence class in this case is the union of the two coordinate axes \(x_1 = 0\) and \(x_2 = 0\), which represent the lines where the product \(x_1 x_2 = 0\).

#### Drawing the Equivalence Classes:
1. For \(y_1 y_2 = c > 0\), the hyperbolas will be in the first and third quadrants.
2. For \(y_1 y_2 = c < 0\), the hyperbolas will be in the second and fourth quadrants.
3. For \(y_1 y_2 = 0\), the equivalence class will be the union of the coordinate axes (lines \(x_1 = 0\) and \(x_2 = 0\)).

Would you like a diagram drawn for these classes or further clarification on any part?

---

### 5 Relative Questions:
1. How can we visualize equivalence relations beyond hyperbolas?
2. Can we find other geometric shapes as equivalence classes for different relations?
3. What happens to the equivalence classes if we restrict the domain to positive real numbers?
4. Can we use other operations (besides multiplication) to define equivalence relations in \(\mathbb{R} \times \mathbb{R}\)?
5. How does the definition of equivalence relations change when applied to higher dimensions?

### Tip:
Always verify equivalence relations by explicitly checking reflexivity, symmetry, and transitivity for clarity.

Define a relation ∼ on R × R as (x1, x2) ∼ (y1, y2) if x1x2 = y1y2.
(i) Check that ∼ is an equivalence relation.
(ii) Identify and draw the equivalence classes.

Explore the equivalence relation defined on R×R by the equation x1 * x2 = y1 * y2. This problem checks the reflexivity, symmetry, and transitivity of the relation and identifies the geometric representation of equivalence classes as hyperbolas and coordinate axes. Suitable for undergraduate students studying algebra and geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation