Let R be an equivalence relation on a set A  . Prove  b € [a] = = [a] = [b]

Let \( R \) be an equivalence relation on a set \( A \), which means that \( R \) satisfies three properties:

1. **Reflexivity:** \( aRa \) for all \( a \in A \).
2. **Symmetry:** If \( aRb \), then \( bRa \).
3. **Transitivity:** If \( aRb \) and \( bRc \), then \( aRc \).

Let \( [a] \) represent the equivalence class of \( a \in A \), i.e.,

\[
[a] = \{ x \in A \mid aRx \}.
\]

### Proof that \( b \in [a] \iff [a] = [b] \):

**(1) Assume \( b \in [a] \):**

This means that \( aRb \), by the definition of the equivalence class. We want to show that \( [a] = [b] \).

- Take any \( x \in [a] \). By definition, \( aRx \). Since \( R \) is symmetric, we know \( aRb \) implies \( bRa \). 
- Since \( R \) is also transitive, from \( bRa \) and \( aRx \), it follows that \( bRx \). Hence, \( x \in [b] \).

Thus, we have shown that \( [a] \subseteq [b] \).

- Now, take any \( y \in [b] \). By definition, \( bRy \). Since \( aRb \) and \( R \) is symmetric, \( bRa \), and by transitivity, from \( aRb \) and \( bRy \), we get \( aRy \). Hence, \( y \in [a] \).

Thus, \( [b] \subseteq [a] \).

Since we have both \( [a] \subseteq [b] \) and \( [b] \subseteq [a] \), it follows that \( [a] = [b] \).

**(2) Assume \( [a] = [b] \):**

If \( [a] = [b] \), then by definition of equivalence classes, \( b \in [a] \).

### Conclusion:

We have shown that \( b \in [a] \iff [a] = [b] \). Therefore, the statement \( b \in [a] \iff [a] = [b] \) is true.

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Would you like more details or have any questions? Here are some related questions:

1. What are some examples of equivalence relations in everyday life?
2. How does an equivalence class partition a set?
3. Can an element belong to more than one equivalence class?
4. What are the implications of transitivity in equivalence relations?
5. How can you apply equivalence relations in set theory and logic?

**Tip:** Equivalence classes group elements of a set that are indistinguishable under a given relation, essentially forming partitions of the set!

Let R be an equivalence relation on a set A. Prove b € [a] == [a] = [b].

Explore the proof that if b belongs to the equivalence class of a, then the equivalence classes of a and b are equal. This problem involves the key concepts of reflexivity, symmetry, and transitivity in equivalence relations and is suitable for advanced high school or undergraduate students.

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