Which of the following is an equivalence relation?

Options

A. is father of
B. is less than
C. is congruent to
D. is an uncle of

Answer

To determine which of the given relations is an equivalence relation, we need to check each relation against the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Understand the Properties of Equivalence Relations**:
- **Reflexive**: For every element \( a \) in set \( A \), the relation must include \( (a, a) \).
- **Symmetric**: For any elements \( a \) and \( b \) in set \( A \), if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation.
- **Transitive**: For any elements \( a, b, c \) in set \( A \), if \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation.

2. **Check Each Relation**:
- **Relation 1: "is a father of"**:
- Reflexive: Not reflexive because no one can be their own father.
- Symmetric: Not symmetric because if \( A \) is the father of \( B \), then \( B \) cannot be the father of \( A \).
- Transitive: Not transitive because if \( A \) is the father of \( B \) and \( B \) is the father of \( C \), then \( A \) is not the father of \( C \).
- Conclusion: This relation is **not an equivalence relation**.

- **Relation 2: "is less than"**:
- Reflexive: Not reflexive because no number is less than itself.
- Symmetric: Not symmetric because if \( A < B \), then \( B < A \) cannot be true.
- Transitive: Transitive because if \( A < B \) and \( B < C \), then \( A < C \).
- Conclusion: This relation is **not an equivalence relation**.

- **Relation 3: "is congruent to"**:
- Reflexive: Reflexive because any figure is congruent to itself.
- Symmetric: Symmetric because if figure \( A \) is congruent to figure \( B \), then \( B \) is congruent to \( A \).
- Transitive: Transitive because if \( A \) is congruent to \( B \) and \( B \) is congruent to \( C \), then \( A \) is congruent to \( C \).
- Conclusion: This relation **is an equivalence relation**.

- **Relation 4: "is an uncle of"**:
- Reflexive: Not reflexive because no one can be their own uncle.
- Symmetric: Not symmetric because if \( A \) is an uncle of \( B \), then \( B \) cannot be an uncle of \( A \).
- Transitive: Not transitive because if \( A \) is an uncle of \( B \) and \( B \) is an uncle of \( C \), it does not imply that \( A \) is an uncle of \( C \).
- Conclusion: This relation is **not an equivalence relation**.

3. **Final Conclusion**:
- The only relation that satisfies all three properties (reflexive, symmetric, and transitive) is "is congruent to". Therefore, it is the equivalence relation.