To solve this, we need to calculate the number of ordered pairs in the equivalence relation on set $S = \{1, 2, 3, 4, 5, 6\}$ induced by the partition $P = \{\{1, 4\}, \{2, 5\}, \{3, 6\}\}$.

Step-by-Step Solution:

1. Understand the Partition: Each subset in $P$ (i.e., $\{1, 4\}$, $\{2, 5\}$, and $\{3, 6\}$) represents an equivalence class. Elements within each subset are considered equivalent to each other.

2. Counting Ordered Pairs per Equivalence Class:

   • For any two elements that are equivalent, they will form pairs in both directions (e.g., if 1 is equivalent to 4, both $(1, 4)$ and $(4, 1)$ are in the equivalence relation, plus the reflexive pairs like $(1, 1)$ and $(4, 4)$).
   • For each set $\{a, b\}$, the ordered pairs are: $(a, a), (a, b), (b, a), (b, b)$
   • This gives $4$ ordered pairs per equivalence class.

3. Total Number of Ordered Pairs:

   • Since there are three equivalence classes and each produces $4$ ordered pairs, the total is: $4 \times 3 = 12$

Answer:

The correct answer is: $\text{12}$

Would you like a further breakdown of any part of this explanation, or have any additional questions?

Here are 5 related questions you might find helpful to expand on this concept:

1. What is an equivalence relation, and how is it defined on a set?
2. How would the answer change if one of the equivalence classes had three elements instead of two?
3. What properties must a relation satisfy to be an equivalence relation?
4. How can we determine the number of ordered pairs in the equivalence relation if the partitioning subsets vary in size?
5. Can an equivalence relation have only reflexive pairs? If yes, under what conditions?

Tip: When dealing with equivalence relations and partitions, remember that elements within each subset of the partition are all related to each other.