Equivalence Relation and Partition:

The equivalence relation is defined by the partition $\{\{1,2\}, \{3,4\}\}$. A partition divides a set into subsets, and in this case, we have two subsets: $\{1, 2\}$ and $\{3, 4\}$. The equivalence relation will identify the elements within each subset as equivalent.

Equivalence Classes:

1. Equivalence Class [1]:
   Since $\{1, 2\}$ is a subset in the partition, the equivalence class $[1]$ will consist of all elements that are related to 1. This includes both 1 and 2 (because they belong to the same subset).
   $[1] = \{1, 2\}$

2. Equivalence Class [2]:
   Similarly, the equivalence class $[2]$ will also consist of 1 and 2, since they are in the same subset.
   $[2] = \{1, 2\}$

3. Equivalence Class [3]:
   Since $\{3, 4\}$ is another subset in the partition, the equivalence class $[3]$ will consist of both 3 and 4.
   $[3] = \{3, 4\}$

4. Equivalence Class [4]:
   Likewise, the equivalence class $[4]$ will consist of both 3 and 4.
   $[4] = \{3, 4\}$

Members of the Equivalence Relation:

The equivalence relation is formed by pairs $(a, b)$ where $a$ and $b$ belong to the same equivalence class. Therefore, the equivalence relation is:

$\{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4)\}$

These are all the pairs of elements that are equivalent under the relation, corresponding to the subsets $\{1, 2\}$ and $\{3, 4\}$.

Summary:

• Equivalence Relation Members: $\{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4)\}$

• Equivalence Classes:

  • $[1] = \{1, 2\}$
  • $[2] = \{1, 2\}$
  • $[3] = \{3, 4\}$
  • $[4] = \{3, 4\}$

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Would you like further details on equivalence relations or partitions? Here are some related questions to explore:

1. How do you prove that a relation is an equivalence relation?
2. What is the difference between an equivalence relation and a partial order?
3. Can an equivalence relation be defined on a set with more subsets?
4. How would the equivalence relation change if the partition were $\{\{1, 3\}, \{2, 4\}\}$?
5. What are some applications of equivalence relations in mathematics?

Tip: Equivalence relations are key in abstract algebra and set theory, helping group elements based on their shared properties. They help form structures like quotient sets and quotient spaces.