Given the relation $R = \{ (x, y) \mid x - y = 0 \} \subseteq \mathbb{R} \times \mathbb{R}$, which can also be written as $R = \{ (x, y) \mid x = y \}$, let's analyze the truth of each statement:

1. $R$ is a transitive relation: A relation is transitive if for any $x, y, z$, whenever $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$.

   • In this case, if $(x, y) \in R$ implies $x = y$, and if $(y, z) \in R$ implies $y = z$, then clearly $x = z$. So, $R$ is transitive.

2. $R$ is a function: A relation is a function if for every $x$, there is exactly one $y$ such that $(x, y) \in R$.

   • In this case, the relation says $x = y$, meaning that for each $x$, there is only one $y$, which is $y = x$. Thus, $R$ behaves like the identity function. So, $R$ is a function.

3. $R$ is not an equivalence relation: To be an equivalence relation, a relation must be reflexive, symmetric, and transitive.

   • We have already established that $R$ is transitive.
   • As we will show below, $R$ is reflexive and symmetric as well. Therefore, $R$ is an equivalence relation, so this statement is false.

4. $R$ is a reflexive relation: A relation is reflexive if for every $x \in \mathbb{R}$, $(x, x) \in R$.

   • In this case, for every $x$, $x - x = 0$, so $(x, x) \in R$. Thus, $R$ is reflexive.

5. $R$ is a symmetric relation: A relation is symmetric if whenever $(x, y) \in R$, it follows that $(y, x) \in R$.

   • Here, if $(x, y) \in R$, we have $x = y$, which means $(y, x) \in R$ as well. So, $R$ is symmetric.

Conclusion:

• $R$ is transitive.
• $R$ is a function.
• $R$ is an equivalence relation (so the statement that it isn't is false).
• $R$ is reflexive.
• $R$ is symmetric.

Do you want further details or explanations?

5 Related Questions:

1. What conditions must be met for a relation to be an equivalence relation?
2. Can a relation be reflexive but not symmetric or transitive?
3. What is the difference between a relation and a function in mathematics?
4. How would the relation $R = \{ (x, y) \mid x + y = 0 \}$ differ from the one given?
5. What are examples of relations that are transitive but not reflexive?

Tip:

When analyzing relations, always check the three key properties—reflexivity, symmetry, and transitivity—to determine if it is an equivalence relation.