To prove that the given relation $(x_1, y_1) \sim (x_2, y_2)$, defined on $\mathbb{R}^2$ by the condition $x_1 + y_1 = x_2 + y_2$, is an equivalence relation, we must verify the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

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1. Reflexivity

A relation is reflexive if $(x_1, y_1) \sim (x_1, y_1)$ for all $(x_1, y_1) \in \mathbb{R}^2$.

• From the definition, $(x_1, y_1) \sim (x_1, y_1)$ means $x_1 + y_1 = x_1 + y_1$.
• This is clearly true because equality holds for any $(x_1, y_1)$.

Thus, the relation is reflexive.

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2. Symmetry

A relation is symmetric if $(x_1, y_1) \sim (x_2, y_2)$ implies $(x_2, y_2) \sim (x_1, y_1)$.

• From the definition, $(x_1, y_1) \sim (x_2, y_2)$ means $x_1 + y_1 = x_2 + y_2$.
• By the symmetry of equality, $x_2 + y_2 = x_1 + y_1$ also holds.
• Therefore, if $(x_1, y_1) \sim (x_2, y_2)$, it follows that $(x_2, y_2) \sim (x_1, y_1)$.

Thus, the relation is symmetric.

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3. Transitivity

A relation is transitive if $(x_1, y_1) \sim (x_2, y_2)$ and $(x_2, y_2) \sim (x_3, y_3)$ together imply $(x_1, y_1) \sim (x_3, y_3)$.

• From the definition:
  • $(x_1, y_1) \sim (x_2, y_2)$ implies $x_1 + y_1 = x_2 + y_2$.
  • $(x_2, y_2) \sim (x_3, y_3)$ implies $x_2 + y_2 = x_3 + y_3$.
• From these two equalities, $x_1 + y_1 = x_3 + y_3$ follows by the transitivity of equality.
• Hence, $(x_1, y_1) \sim (x_3, y_3)$.

Thus, the relation is transitive.

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Conclusion

Since the relation satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

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Would you like me to expand on any part of the proof or provide examples? 😊

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Here are 5 related questions to deepen understanding:

1. What are equivalence classes of the relation $(x_1, y_1) \sim (x_2, y_2)$ under this equivalence?
2. How can this equivalence relation be applied in geometry or algebra?
3. Is the sum $x + y$ in the equivalence relation invariant under certain transformations?
4. Can you define another equivalence relation on $\mathbb{R}^2$ based on different operations (e.g., multiplication)?
5. How would this relation change if we replaced $x_1 + y_1 = x_2 + y_2$ with $x_1^2 + y_1^2 = x_2^2 + y_2^2$?

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Tip: Always verify the properties of an equivalence relation (reflexivity, symmetry, transitivity) systematically to ensure clarity in proofs.