Suppose R is a relation on any set A. Prove that if R is symmetric and transitive

then it is an equivalent relation

To prove that if \( R \) is a relation on a set \( A \) and is symmetric and transitive, then \( R \) is an equivalence relation, we need to show that \( R \) also satisfies the property of reflexivity. An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity.

### Given:
- \( R \) is a relation on \( A \).
- \( R \) is **symmetric**: If \( (a, b) \in R \), then \( (b, a) \in R \).
- \( R \) is **transitive**: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

### To Prove:
- \( R \) is **reflexive**: For every \( a \in A \), \( (a, a) \in R \).

If we can prove that \( R \) is reflexive, then we can conclude that \( R \) is an equivalence relation because it will satisfy all three required properties.

### Proof

1. **Consider any element \( a \in A \).** We want to show that \( (a, a) \in R \).

2. **Since \( R \) is symmetric and transitive,** let us examine the implications:
   - Suppose there exists some \( b \in A \) such that \( (a, b) \in R \) (or if not, \( R \) is empty, in which case reflexivity does not apply since there are no elements in \( A \)).

3. If \( R \)

Suppose R is a relation on any set A. Prove that if R is symmetric and transitive, then it is an equivalence relation.

Explore a proof showing that a symmetric and transitive relation on any set is an equivalence relation. This step-by-step solution covers essential properties of relations, such as symmetry, transitivity, and reflexivity, and is suitable for undergraduate students studying advanced algebra.

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