Determine whether the relation R on the set of all real numbers is reflexive, symmetric, 

antisymmetric, and/or transitive, where (𝑥, 𝑦) ∈ 𝑅 if and only if

(d) 𝑥 is a multiple of 𝑦

Let's analyze the relation \( R \) on the set of all real numbers \( \mathbb{R} \), where \( (x, y) \in R \) if and only if \( x \) is a multiple of \( y \). Specifically, \( x \) is a multiple of \( y \) if there exists a real number \( k \) such that \( x = k \cdot y \).

### Reflexive
A relation is **reflexive** if every element is related to itself, i.e., \( (x, x) \in R \) for all \( x \in \mathbb{R} \).

- **Analysis**: For any real number \( x \), \( x = 1 \cdot x \). Hence, \( x \) is always a multiple of itself, so \( (x, x) \in R \) for all \( x \in \mathbb{R} \).

**Conclusion**: The relation \( R \) is **reflexive**.

### Symmetric
A relation is **symmetric** if whenever \( (x, y) \in R \), then \( (y, x) \in R \) as well.

- **Analysis**: If \( (x, y) \in R \), then \( x = k \cdot y \) for some real number \( k \). For \( (y, x) \) to be in \( R \), \( y \) must be a multiple of \( x \), meaning there must exist a real number \( m \) such that \( y = m \cdot x \).

  - If \( k = 0 \), then \( x = 0 \), and \( (0, y) \) is in \( R \) for any \( y \), but \( (y, 0) \) will only be in \( R \) if \( y = 0 \), so symmetry may not hold.
  - If \( k \neq 0 \), then symmetry does not necessarily hold because there is no guarantee that \( y = m \cdot x \) (i.e., that \( y \) is a multiple of \( x \)).

**Conclusion**: The relation \( R \) is **not symmetric**.

### Antisymmetric
A relation is **antisymmetric** if whenever \( (x, y) \in R \) and \( (y, x) \in R \), then \( x = y \).

- **Analysis**: If \( (x, y) \in R \) and \( (y, x) \in R \), then \( x \) is a multiple of \( y \) and \( y \) is a multiple of \( x \). This implies that \( x = k \cdot y \) and \( y = m \cdot x \), leading to \( x = k \cdot m \cdot x \). For this to be true for all \( x \), \( k \cdot m = 1 \). The only real numbers that satisfy this condition for all \( x \) are \( k = 1 \) and \( m = 1 \), which implies \( x = y \).

**Conclusion**: The relation \( R \) is **antisymmetric**.

### Transitive
A relation is **transitive** if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).

- **Analysis**: If \( (x, y) \in R \), then \( x = k \cdot y \) for some \( k \), and if \( (y, z) \in R \), then \( y = m \cdot z \) for some \( m \). Substituting, \( x = k \cdot (m \cdot z) = (k \cdot m) \cdot z \), which implies \( x \) is a multiple of \( z \), so \( (x, z) \in R \).

**Conclusion**: The relation \( R \) is **transitive**.

### Final Summary:
- **Reflexive**: Yes
- **Symmetric**: No
- **Antisymmetric**: Yes
- **Transitive**: Yes

Would you like more details on any of these properties, or do you have any other questions?

Here are five related questions you might find interesting:
1. How would the properties of this relation change if we restricted the set to positive integers?
2. Can you think of a real-world scenario where a similar relation might apply?
3. What happens if we consider the relation where \( y \) is a multiple of \( x \) instead?
4. How do these properties differ when considering the set of nonzero real numbers?
5. What are some examples of relations that are symmetric but not antisymmetric?

**Tip**: When analyzing the symmetry of a relation, always consider edge cases, like when the elements involved might be zero, as they can often break symmetry.

Explore the properties of relation R on the set of real numbers. Understand if R is reflexive, symmetric, antisymmetric, and/or transitive through detailed analysis and examples. Suitable for undergraduate level mathematics.

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