To determine the equivalence classes $E(x)$ for each element $x$ in $A$ with respect to the equivalence relation $R$, we need to analyze the pairs in $R$ and group the elements that are equivalent.

Given Relation $R$:

$R = \{(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5), (2, 2), (2, 6), (6, 2), (6, 6), (4, 4)\}$

An equivalence relation satisfies reflexivity, symmetry, and transitivity. Using this relation, let's find the equivalence classes $E(x)$.

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Steps:

1. Find $E(1)$:

   • Look for all $y$ such that $(1, y) \in R$ or $(y, 1) \in R$.
   • From $R$: $(1, 1), (1, 3), (1, 5), (3, 1), (5, 1)$.
   • Thus, $1, 3,$ and $5$ are related.
   • Equivalence class $E(1) = \{1, 3, 5\}$.

2. Find $E(2)$:

   • Look for all $y$ such that $(2, y) \in R$ or $(y, 2) \in R$.
   • From $R$: $(2, 2), (2, 6), (6, 2)$.
   • Thus, $2$ and $6$ are related.
   • Equivalence class ( E(2) = {2, 6}.

3. Find $E(3)$:

   • Look for all $y$ such that $(3, y) \in R$ or $(y, 3) \in R$.
   • From $R$: $(3, 1), (3, 3), (3, 5), (1, 3), (5, 3)$.
   • Thus, $1, 3,$ and $5$ are related.
   • Equivalence class $E(3) = \{1, 3, 5\}$.

4. Find $E(4)$:

   • Look for all $y$ such that $(4, y) \in R$ or $(y, 4) \in R$.
   • From $R$: $(4, 4)$.
   • Thus, $4$ is only related to itself.
   • Equivalence class ( E(4) = {4}.

5. Find $E(5)$:

   • Look for all $y$ such that $(5, y) \in R$ or $(y, 5) \in R$.
   • From $R$: $(5, 1), (5, 3), (5, 5), (1, 5), (3, 5)$.
   • Thus, $1, 3,$ and $5$ are related.
   • Equivalence class $E(5) = \{1, 3, 5\}$.

6. Find $E(6)$:

   • Look for all $y$ such that $(6, y) \in R$ or $(y, 6) \in R$.
   • From $R$: $(6, 2), (6, 6), (2, 6)$.
   • Thus, $2$ and $6$ are related.
   • Equivalence class ( E(6) = {2, 6}.

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Final Results:

• $E(1) = \{1, 3, 5\}$
• $E(2) = \{2, 6\}$
• $E(3) = \{1, 3, 5\}$
• $E(4) = \{4\}$
• $E(5) = \{1, 3, 5\}$
• $E(6) = \{2, 6\}$

Would you like more details on how these equivalence classes were derived or any additional examples?

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5 Related Questions:

1. How do equivalence relations differ from general binary relations?
2. What are the key properties of equivalence relations?
3. How would $R$ change if we added new elements to $A$?
4. What would happen if reflexivity, symmetry, or transitivity failed in $R$?
5. Can equivalence classes overlap in any relation?

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Tip:

When analyzing equivalence relations, always verify the reflexivity, symmetry, and transitivity properties before determining equivalence classes.