1. Key Concepts and Formulas

• Symmetric Relation: A relation $R$ on a set $S$ is symmetric if for every $(x, y) \in R$, it is true that $(y, x) \in R$.
• Transitive Relation: A relation $R$ on a set $S$ is transitive if for every $(x, y) \in R$ and $(y, z) \in R$, it is true that $(x, z) \in R$.
• Universal Relation: For a set $S$, the universal relation $S \times S$ contains all possible ordered pairs of elements from $S$.

2. Step-by-Step Solution

We are given the set $S = \{a, b, c\}$ and the initial relation $R_0 = \{(a, b), (b, c)\}$. We need to add the minimum number of elements to $R_0$ to make it both symmetric and transitive.

Step 1: Enforce Symmetry To make the relation symmetric, for every pair $(x, y)$ in the relation, the pair $(y, x)$ must also be present.

• Since $(a, b) \in R_0$, we must add $(b, a)$.
• Since $(b, c) \in R_0$, we must add $(c, b)$.

Let $R_1$ be the relation after adding these elements for symmetry: $R_1 = R_0 \cup \{(b, a), (c, b)\} = \{(a, b), (b, c), (b, a), (c, b)\}$ The elements added in this step are $(b, a)$ and $(c, b)$. The number of elements in $R_1$ is 4.

Step 2: Enforce Transitivity and Maintain Symmetry Now we need to ensure transitivity for $R_1$. For any $(x, y) \in R_1$ and $(y, z) \in R_1$, we must have $(x, z) \in R_1$. As we add elements for transitivity, we must also ensure their symmetric counterparts are added to maintain symmetry.

Let's examine all possible transitive implications from $R_1$:

• From $(a, b) \in R_1$ and $(b, c) \in R_1$, we require $(a, c)$.
  • To maintain symmetry, we must also add $(c, a)$.
  • New elements to add: $(a, c)$ and $(c, a)$.
• From $(b, c) \in R_1$ and $(c, b) \in R_1$, we require $(b, b)$.
  • This is a reflexive pair and is symmetric by itself.
  • New element to add: $(b, b)$.
• From $(a, b) \in R_1$ and $(b, a) \in R_1$, we require $(a, a)$.
  • This is a reflexive pair and is symmetric by itself.
  • New element to add: $(a, a)$.
• From $(c, b) \in R_1$ and $(b, a) \in R_1$, we require $(c, a)$. (Already identified above).
• From $(b, a) \in R_1$ and $(a, b) \in R_1$, we require $(b, b)$. (Already identified above).
• From $(c, b) \in R_1$ and $(b, c) \in R_1$, we require $(c, c)$.
  • This is a reflexive pair and is symmetric by itself.
  • New element to add: $(c, c)$.

The new unique elements that must be added to $R_1$ are: $(a, c)$, $(c, a)$, $(b, b)$, $(a, a)$, and $(c, c)$.

Let $R_2$ be the relation after adding these elements: $R_2 = R_1 \cup \{(a, c), (c, a), (b, b), (a, a), (c, c)\}$ $R_2 = \{(a, b), (b, c), (b, a), (c, b), (a, c), (c, a), (b, b), (a, a), (c, c)\}$ The number of elements added in this step is 5. The total number of elements in $R_2$ is $4 + 5 = 9$.

Step 3: Verify the Final Relation The set $S = \{a, b, c\}$ has $3 \times 3 = 9$ possible ordered pairs. The relation $R_2$ contains all 9 possible ordered pairs, which means $R_2 = S \times S$.

• The universal relation $S \times S$ is always symmetric, because if $(x, y) \in S \times S$, then $x, y \in S$, so $(y, x)$ is also in $S \times S$.
• The universal relation $S \times S$ is always transitive, because if $(x, y) \in S \times S$ and $(y, z) \in S \times S$, then $x, y, z \in S$, so $(x, z)$ is also in $S \times S$.

Therefore, $R_2$ is both symmetric and transitive.

Step 4: Calculate the Minimum Number of Elements Added The initial relation $R_0$ had 2 elements. The final symmetric and transitive relation $R_2$ has 9 elements. The minimum number of elements that must be added is the difference: $\text{Number of elements added} = (\text{Number of elements in } R_2) - (\text{Number of elements in } R_0)$ $\text{Number of elements added} = 9 - 2 = 7$

3. Common Mistakes & Tips

• Forgetting Symmetry When Enforcing Transitivity: When you add a pair $(x, z)$ for transitivity, remember to also add its symmetric counterpart $(z, x)$ if it's not already present, to ensure the relation remains symmetric.
• Double Counting: Systematically list the elements added at each stage to avoid counting the same element multiple times.
• Recognizing the Universal Relation: If the initial relation connects all elements of the set (e.g., $a$ to $b$, $b$ to $c$), and you need symmetry and transitivity, the resulting relation is often the universal relation $S \times S$. This simplifies the final count.

4. Summary

We started with the relation $R = \{(a, b), (b, c)\}$ on the set $\{a, b, c\}$. To achieve symmetry, we added $(b, a)$ and $(c, b)$. Then, to ensure transitivity while maintaining symmetry, we systematically identified and added required pairs, including reflexive pairs like $(a, a), (b, b), (c, c)$ and cross-pairs like $(a, c), (c, a)$. This process led to the universal relation $S \times S$, which contains all $3 \times 3 = 9$ possible ordered pairs. Since the initial relation had 2 elements, the minimum number of elements that had to be added is $9 - 2 = 7$.

The final answer is $\boxed{7}$.