To solve the problem, we need to compute the membership functions for the fuzzy set operations using the given membership functions:

• $\mu_A(x) = \frac{x}{x+1}$
• $\mu_B(x) = \frac{1}{x^2 + 10}$
• $\mu_C(x) = \frac{1}{10^x}$

We will compute the membership functions for the following fuzzy set operations:

a) $A \cup B$, $B \cap C$

• Union: For fuzzy sets, the membership function of $A \cup B$ is given by: $\mu_{A \cup B}(x) = \max(\mu_A(x), \mu_B(x))$
• Intersection: For fuzzy sets, the membership function of $B \cap C$ is given by: $\mu_{B \cap C}(x) = \min(\mu_B(x), \mu_C(x))$

b) $A \cup B \cup C$, $A \cap B \cap C$

• Union: The membership function of $A \cup B \cup C$ is: $\mu_{A \cup B \cup C}(x) = \max(\mu_A(x), \mu_B(x), \mu_C(x))$
• Intersection: The membership function of $A \cap B \cap C$ is: $\mu_{A \cap B \cap C}(x) = \min(\mu_A(x), \mu_B(x), \mu_C(x))$

c) $A \cap \overline{C}$, $\overline{B} \cup C$

• Complement: The membership function of $\overline{C}$ is given by: $\mu_{\overline{C}}(x) = 1 - \mu_C(x)$ Similarly, the membership function of $\overline{B}$ is: $\mu_{\overline{B}}(x) = 1 - \mu_B(x)$
• Intersection: The membership function of $A \cap \overline{C}$ is: $\mu_{A \cap \overline{C}}(x) = \min(\mu_A(x), 1 - \mu_C(x))$
• Union: The membership function of $\overline{B} \cup C$ is: $\mu_{\overline{B} \cup C}(x) = \max(1 - \mu_B(x), \mu_C(x))$

Next Steps:

Would you like me to plot the graphs of these membership functions for specific $x$-values or provide the detailed calculations for a few of these set operations?

Here are some related questions:

1. Would you like to see more examples of fuzzy set operations?
2. Should I explain more about the properties of union and intersection in fuzzy sets?
3. Do you need a specific range of $x$ values for plotting the functions?
4. Are you familiar with other forms of set operations in fuzzy logic, such as complement or difference?
5. Would you like to explore fuzzy set theory applications in real-world problems?

Tip: Fuzzy set theory is powerful for modeling uncertain or imprecise information, often used in decision-making and control systems!