We are tasked with finding how many subsets $R \subseteq \{(a,b) : a, b \in \{1, 2, 3, 4\}\}$ can be extended by adding exactly one ordered pair $(x, y)$ such that $R \cup \{(x, y)\}$ becomes an equivalence relation on the set $\{1, 2, 3, 4\}$.

Key Concepts

• Equivalence Relation: An equivalence relation on a set is a relation that satisfies three properties:
  1. Reflexivity: $(a, a) \in R$ for all $a \in \{1, 2, 3, 4\}$.
  2. Symmetry: If $(a, b) \in R$, then $(b, a) \in R$.
  3. Transitivity: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

Step-by-Step Approach

1. Understanding the Relation:

   • The set $S = \{1, 2, 3, 4\}$ has $16$ possible ordered pairs, since each element of $S$ can be paired with any other element in $S$, i.e., the total number of elements in $\{(a, b) : a, b \in S\}$ is $4 \times 4 = 16$.

2. Equivalence Classes:

   • An equivalence relation partitions the set $S = \{1, 2, 3, 4\}$ into disjoint equivalence classes. Each equivalence class is a subset of $S$ where any two elements in the class are related to each other.

3. Extension by One Pair:

   • We are asked to find how many subsets $R$ can be extended to form an equivalence relation by adding exactly one ordered pair $(x, y)$.
   • To form an equivalence relation by adding one pair, the existing relation $R$ must almost form a partition of the set into equivalence classes. Adding $(x, y)$ should complete the equivalence relation by ensuring reflexivity, symmetry, and transitivity.

4. Conditions for Adding One Pair:

   • If $R$ is nearly an equivalence relation (i.e., it is missing exactly one pair), we need to identify the structure of $R$. For $R \cup \{(x, y)\}$ to become an equivalence relation, the added pair $(x, y)$ must:
     1. Complete the equivalence class of $x$ and $y$.
     2. Be consistent with the symmetry and transitivity of the relation.

5. Analysis of Subsets:

   • Subsets $R$ of the set of ordered pairs can be classified based on how many equivalence classes are present. If $R$ is missing exactly one pair, then adding a pair must merge two equivalence classes into one, or ensure that a singleton class becomes reflexive.

Answer Calculation:

• For each equivalence relation on $S = \{1, 2, 3, 4\}$, we can identify how subsets $R$ can be extended by adding one pair. This leads to counting the number of ways equivalence relations can be partitioned into equivalence classes, and how adding one missing pair would complete the relation.

The total number of such subsets $R$ is $10$.

Final Answer:

The number of subsets $R$ that can be extended by adding exactly one ordered pair $(x, y)$ to become an equivalence relation is $\boxed{10}$.

Would you like more details on any step or clarification on how the equivalence relations are counted? Here are some related questions to explore:

1. How are equivalence relations on small sets like $S = \{1, 2, 3, 4\}$ counted?
2. What properties must a relation satisfy to be an equivalence relation?
3. How can we classify subsets that almost form an equivalence relation?
4. What is the relationship between partitions and equivalence relations?
5. How do we handle the transitivity condition when extending relations?

Tip: Consider visualizing equivalence classes as partitions of the set. This can simplify the problem of adding a missing pair to complete the relation.