nsider the relation
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R={(x,y)∣x−y=0}⊂R×R on the set
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R. Which of the following is/are true?

R
R is a transitive relation.

R
R is a function.

R
R is not an equivalence relation.

R
R is a reflexive relation.

R
R is a symmetric relation.

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.

Consider the relation R = {(x, y) | x - y = 0} ⊂ R × R on the set R. Which of the following is/are true? R is a transitive relation. R is a function. R is not an equivalence relation. R is a reflexive relation. R is a symmetric relation.

Explore the properties of the relation R = {(x, y) | x = y} on the set of real numbers. This problem covers the concepts of transitive, reflexive, and symmetric relations and determines whether R is an equivalence relation or a function. Suitable for undergraduate students.

Math Problem Statement

nsider the relation R

{ ( x , y ) ∣ x − y

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