The task is to determine which of the given sets are subsets of the fuzzy set $A$ defined in the universal set $U = \{u_1, u_2, u_3, u_4, u_5\}$.

The fuzzy set $A$ is defined as:

$A = 0.1/u_1 + 0.4/u_2 + 1/u_3 + 0.7/u_4 + 0.9/u_5$

A fuzzy set $B$ is a subset of $A$ if for every element $u_i$ in $U$, the membership value of $u_i$ in $B$ is less than or equal to its membership value in $A$.

Let’s analyze each option:

1. First Option: $0.1/u_1 + 0.4/u_2 + 1/u_3 + 0.7/u_4 + 0.9/u_5$ Comparing with $A$, the membership values for all elements $u_1, u_2, u_3, u_4, u_5$ are identical. Therefore, this set is clearly a subset of $A$.

2. Second Option: $1/u_1 + 0.9/u_2 + 1/u_3 + 0.9/u_4 + 0.5/u_5$ Comparing with $A$, the membership values are:

   • $u_1$: 1 (in this option) is greater than 0.1 (in $A$), so this is not a subset of $A$.

3. Third Option: $0.2/u_1 + 0.5/u_2 + 0.6/u_3 + 0.4/u_4 + 0.8/u_5$ Comparing with $A$, the membership values are:

   • $u_1$: 0.2 (in this option) is greater than 0.1 (in $A$), so this is not a subset of $A$.

4. Fourth Option: $0.4/u_1 + 0.8/u_2 + 1/u_3 + 0.8/u_4 + 0.9/u_5$ Comparing with $A$, the membership values are:

   • $u_1$: 0.4 (in this option) is greater than 0.1 (in $A$), so this is not a subset of $A$.

Conclusion:

• The first option is the only subset of $A$.

Do you need further clarification or details on fuzzy subsets?

Here are 5 related questions to expand on this topic:

1. How do fuzzy sets differ from traditional crisp sets?
2. Can a fuzzy set have multiple subsets with varying degrees of membership?
3. What are some practical applications of fuzzy logic in real life?
4. How would we determine the union and intersection of two fuzzy sets?
5. How does the membership function affect operations like subset, union, and intersection?

Tip: When comparing fuzzy sets, always check each element's membership value individually to ensure the subset condition holds.