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The problem asks whether the symmetric difference of linear orders defined on \( A \), where \( |A| = 7 \), can be an equivalence relation.

### Key points to analyze:
1. **Symmetric Difference**:
   The symmetric difference of two sets \( X \) and \( Y \), denoted \( X \Delta Y \), is defined as:
   \[
   X \Delta Y = (X \setminus Y) \cup (Y \setminus X)
   \]
   In the context of linear orders, this represents the set of pairs where the two orders differ.

2. **Properties of an Equivalence Relation**:
   An equivalence relation must satisfy:
   - Reflexivity: \( x \sim x \) for all \( x \).
   - Symmetry: If \( x \sim y \), then \( y \sim x \).
   - Transitivity: If \( x \sim y \) and \( y \sim z \), then \( x \sim z \).

### Analysis:
1. **Reflexivity**:
   For the symmetric difference to form an equivalence relation, \( X \Delta X \) must hold for all linear orders \( X \). However:
   \[
   X \Delta X = \emptyset
   \]
   Hence, the symmetric difference does not equate \( X \) to itself in a non-trivial way, failing reflexivity.

2. **Symmetry**:
   By definition of symmetric difference, \( X \Delta Y = Y \Delta X \), so symmetry holds.

3. **Transitivity**:
   For the symmetric difference to be transitive:
   \[
   (X \Delta Y) \cap (Y \Delta Z) \subseteq (X \Delta Z)
   \]
   However, this does not generally hold because the symmetric difference is not associative in the manner required for equivalence.

### Conclusion:
The symmetric difference of linear orders on \( A \) cannot be an equivalence relation because it fails **reflexivity** and **transitivity**.

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Would you like me to expand on specific aspects of the problem, such as exploring related equivalence relations or providing examples?  

### Further Questions:
1. What are examples of equivalence relations defined on linear orders?
2. Can symmetric differences form other meaningful structures (e.g., metric spaces)?
3. How would reflexivity fail explicitly in the case of symmetric difference for orders on \( A \)?
4. What does this result imply for equivalence relations in set theory and order theory?
5. Can we generalize the result for other cardinalities of \( A \)?

### Tip:
When testing equivalence relations, always verify the three properties explicitly using specific counterexamples when possible.

Пусть |A| = 7. Может ли симметрическая разность определенных на A линейных порядков (как строгих, так и нестрогих) быть отношением эквивалентности?

Explore whether the symmetric difference of linear orders on a set with cardinality |A| = 7 can form an equivalence relation. This problem delves into set theory and the properties of equivalence relations, such as reflexivity, symmetry, and transitivity.

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