If U is the universal set and `A sube B sube U` then which of the following is true?

Options

A. `U- B = U-A`
B. `U-A sube U - B`
C. `U - B sube U -A`
D. `U - A sube (U-A) cap (U-B)`

Answer

To solve the problem step by step, we need to analyze the relationships between the sets A, B, and U, where U is the universal set.

### Step-by-Step Solution:

1. **Understanding the Universal Set and Subsets**:
- Let \( U \) be the universal set.
- We know that \( A \subseteq B \subseteq U \). This means that all elements of set \( A \) are also in set \( B \), and all elements of set \( B \) are in the universal set \( U \).

2. **Visualizing with a Venn Diagram**:
- Draw a rectangle to represent the universal set \( U \).
- Inside this rectangle, draw a circle for set \( B \).
- Inside the circle for \( B \), draw a smaller circle for set \( A \).
- This visualization helps us understand the relationships: \( A \) is completely contained within \( B \), which is completely contained within \( U \).

3. **Analyzing the Options**:
- We need to evaluate the truth of various statements regarding the relationships between these sets. The options typically involve set differences and subsets.

4. **Evaluating Each Option**:
- **Option 1: \( B \subseteq U' \)** (where \( U' \) is the complement of \( U \)):
- This is false because \( B \) is a subset of \( U \), not its complement.
- **Option 2: \( A \subseteq U' \)**:
- This is also false for the same reason; \( A \) is contained in \( U \).
- **Option 3: \( U - A \subseteq U - B \)**:
- This is true. Since \( A \subseteq B \), removing \( A \) from \( U \) will leave more elements than removing \( B \) from \( U \). Therefore, the complement of \( A \) in \( U \) is a subset of the complement of \( B \) in \( U \).
- **Option 4: \( U - B \subseteq U - A \)**:
- This is false. The complement of \( B \) will not necessarily be a subset of the complement of \( A \) since \( B \) contains more elements than \( A \).

5. **Conclusion**:
- The only true statement among the options is **Option 3: \( U - A \subseteq U - B \)**.