If A is a subset of B, then which one of the following is correct?

Options

A. 1) `A^(c ) sube B^(c )`
B. 2) `B^(c ) sube A^(c )`
C. 3) `A^(c ) = B^(c )`
D. 4) `A sube A cap B`

Answer

To solve the problem, we need to analyze the relationship between sets A and B, given that A is a subset of B. Let's go through the steps to determine which statement is correct.

### Step-by-Step Solution:

1. **Understanding the Definition of Subset**:
- If A is a subset of B (denoted as \( A \subseteq B \)), it means that every element of A is also an element of B.

**Hint**: Remember that a subset means all elements of the first set are contained within the second set.

2. **Visualizing with Venn Diagrams**:
- We can represent this relationship using a Venn diagram. In the diagram, set B is the larger circle, and set A is a smaller circle inside B.

**Hint**: Drawing a Venn diagram can help visualize the relationship between the sets.

3. **Understanding Complements**:
- The complement of a set refers to all the elements in the universal set that are not in the given set.
- If we denote the universal set as U, then:
- The complement of A (denoted as \( A' \)) includes all elements in U that are not in A.
- The complement of B (denoted as \( B' \)) includes all elements in U that are not in B.

**Hint**: Recall that the complement of a set includes everything outside that set.

4. **Analyzing the Complements**:
- Since A is a subset of B, any element not in B (i.e., in \( B' \)) cannot be in A either. Therefore, if an element is in \( B' \), it must also be in \( A' \).
- This means that \( B' \subseteq A' \).

**Hint**: Think about what it means for elements to be outside of both sets.

5. **Conclusion**:
- From the above analysis, we can conclude that the complement of B is a subset of the complement of A, which can be written as:
\[ B' \subseteq A' \]
- Thus, the correct statement is that the complement of A is a subset of the complement of B.

### Final Answer:
The correct statement is: **The complement of B is a subset of the complement of A**.