A = {1, 2, 3} and R be the relation defined as R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)} Is R equivalence?

To determine whether the relation \( R = \{(1,1), (1,2), (2,2), (2,3), (3,3)\} \) is an equivalence relation on the set \( A = \{1, 2, 3\} \), we need to check if \( R \) satisfies the three properties of an equivalence relation: **reflexivity**, **symmetry**, and **transitivity**.

### 1. Reflexivity:
A relation \( R \) on a set \( A \) is **reflexive** if every element in \( A \) is related to itself, i.e., for all \( a \in A \), \( (a, a) \in R \).

For \( A = \{1, 2, 3\} \), we need the pairs \( (1, 1), (2, 2), (3, 3) \) to be in \( R \).
- \( (1, 1) \in R \), which is true.
- \( (2, 2) \in R \), which is true.
- \( (3, 3) \in R \), which is true.

Since all elements of \( A \) are related to themselves, the relation \( R \) is **reflexive**.

### 2. Symmetry:
A relation \( R \) on a set \( A \) is **symmetric** if for all \( a, b \in A \), whenever \( (a, b) \in R \), it must also hold that \( (b, a) \in R \).

Check the pairs in \( R \):
- \( (1, 1) \) is symmetric (since \( (1, 1) = (1, 1) \)).
- \( (1, 2) \in R \), but \( (2, 1) \notin R \), so symmetry is **violated**.
- \( (2, 2) \) is symmetric.
- \( (2, 3) \in R \), but \( (3, 2) \notin R \), so symmetry is **violated**.
- \( (3, 3) \) is symmetric.

Since symmetry is violated (for \( (1, 2) \) and \( (2, 3) \)), the relation \( R \) is **not symmetric**.

### 3. Transitivity:
A relation \( R \) on a set \( A \) is **transitive** if for all \( a, b, c \in A \), whenever \( (a, b) \in R \) and \( (b, c) \in R \), it must also hold that \( (a, c) \in R \).

Check the transitive property for the pairs in \( R \):
- \( (1, 1) \in R \) and \( (1, 2) \in R \). We need \( (1, 2) \in R \), which is true.
- \( (1, 2) \in R \) and \( (2, 3) \in R \). We need \( (1, 3) \in R \), but \( (1, 3) \notin R \), so transitivity is **violated**.
- \( (2, 2) \in R \) and \( (2, 3) \in R \). We need \( (2, 3) \in R \), which is true.
- \( (3, 3) \in R \), so no transitivity check is needed here.

Since the transitivity condition is violated (for \( (1, 2) \) and \( (2, 3) \)), the relation \( R \) is **not transitive**.

### Conclusion:
Since the relation \( R \) is **reflexive** but neither **symmetric** nor **transitive**, it is **not an equivalence relation**.

Would you like more details on any of the properties or steps?

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Here are 5 questions to consider based on this topic:

1. What conditions must a relation satisfy to be transitive?
2. How does the lack of symmetry impact a relation in practical scenarios?
3. Can a relation be reflexive but still not an equivalence relation? Why or why not?
4. What changes could be made to the relation \( R \) to make it symmetric?
5. How is the concept of equivalence relations applied in real-world mathematical problems?

**Tip:** When checking if a relation is symmetric, always verify that if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation.

A = {1, 2, 3} and R be the relation defined as R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)}. Is R equivalence?

Explore whether the relation R = {(1,1), (1,2), (2,2), (2,3), (3,3)} on the set A = {1, 2, 3} is an equivalence relation by checking reflexivity, symmetry, and transitivity. This problem is suitable for high school students studying set theory and relations.

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