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.