To show that the relation $\sim$ defined on the set $X$ is an equivalence relation, we must demonstrate that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

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1. Reflexivity

To show reflexivity, we need to prove that for any $x \in X$, $x \sim x$.

• By definition of the partition $P$, every element $x \in X$ belongs to exactly one subset $A \in P$.
• Since $x \in A$, it is trivially true that $x \sim x$ because $x$ and $x$ belong to the same subset $A \in P$.
  Thus, reflexivity holds.

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2. Symmetry

To show symmetry, we need to prove that if $x \sim y$, then $y \sim x$.

• Suppose $x \sim y$. By definition of $\sim$, this means $x, y \in A$ for some $A \in P$.
• The relation $\sim$ is defined based on the subsets of $P$, so if $x, y \in A$, then $y, x \in A$ as well (set membership is symmetric).
  Thus, symmetry holds.

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3. Transitivity

To show transitivity, we need to prove that if $x \sim y$ and $y \sim z$, then $x \sim z$.

• Suppose $x \sim y$ and $y \sim z$. By definition of $\sim$, $x, y \in A_1$ for some $A_1 \in P$, and $y, z \in A_2$ for some $A_2 \in P$.
• Since $P$ is a partition, every element of $X$ belongs to exactly one subset of $P$. Thus, $A_1 = A_2$, meaning $x, y, z \in A_1$.
  Thus, transitivity holds.

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Conclusion

Since $\sim$ satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on $X$.

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Would you like further clarification or additional examples?
Here are five related questions to deepen your understanding:

1. How does the definition of a partition ensure that $\sim$ is well-defined?
2. Can you provide examples of partitions of a set $X$ and the corresponding equivalence relations?
3. Why is set membership inherently symmetric, and how does this relate to symmetry of $\sim$?
4. How does the uniqueness of subsets in a partition guarantee transitivity of $\sim$?
5. What is the difference between an equivalence class and a subset in the context of partitions?

Tip: When verifying equivalence relations, focus on the definitions of reflexivity, symmetry, and transitivity individually to ensure clarity and correctness.