Let A,B and c be three sets . If `AsubBandB in C`, is it true that `AsubC` ? If not , given an example.

Answer

To determine whether \( A \subseteq C \) given that \( A \subseteq B \) and \( B \in C \), we can analyze the definitions of subsets and set membership.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- A set \( A \) is a subset of a set \( B \) (denoted \( A \subseteq B \)) if every element of \( A \) is also an element of \( B \).
- A set \( B \) is an element of a set \( C \) (denoted \( B \in C \)) means that \( B \) is one of the members contained within the set \( C \).

2. **Given Conditions**:
- We know that \( A \subseteq B \).
- We also know that \( B \in C \).

3. **Analyzing the Relationship**:
- From \( A \subseteq B \), it follows that all elements of \( A \) are contained in \( B \).
- However, \( B \in C \) does not imply that all elements of \( B \) are in \( C \). Instead, it means that \( B \) itself is a member of \( C \).

4. **Conclusion**:
- Since \( B \) is an element of \( C \) and not necessarily a subset of \( C \), we cannot conclude that \( A \subseteq C \).
- Therefore, it is not necessarily true that \( A \subseteq C \).

5. **Example to Illustrate**:
- Let’s take an example:
- Let \( A = \{1\} \)
- Let \( B = \{1, 2\} \)
- Let \( C = \{\{1, 2\}, 3\} \)
- Here, \( A \subseteq B \) (since 1 is in \( B \)), and \( B \in C \) (since \( B \) is one of the elements in \( C \)).
- However, \( A \) is not a subset of \( C \) because the only element of \( A \) (which is 1) is not in \( C \).

### Final Answer:
No, it is not true that \( A \subseteq C \) given \( A \subseteq B \) and \( B \in C \). An example is \( A = \{1\}, B = \{1, 2\}, C = \{\{1, 2\}, 3\} \).