If `A sub B "and " C, " then " A in C ` is this statement true ?

Explanation

To determine whether the statement "If \( A \subset B \) and \( B \subset C \), then \( A \in C \)" is true or false, we need to analyze the concepts of subsets and membership in set theory.

### Step-by-Step Solution:

1. **Understanding the Notation**:
- \( A \subset B \) means that all elements of set \( A \) are also elements of set \( B \).
- \( B \subset C \) means that all elements of set \( B \) are also elements of set \( C \).
- \( A \in C \) means that set \( A \) is an element of set \( C \).

2. **Analyzing the Given Information**:
- We have \( A \subset B \) and \( B \subset C \). This implies that every element of \( A \) is in \( B \), and every element of \( B \) is in \( C \).

3. **Transitive Property of Subsets**:
- From the definitions, if \( A \subset B \) and \( B \subset C \), we can conclude that \( A \subset C \). This means all elements of \( A \) are also in \( C \).

4. **Distinguishing Between Subset and Membership**:
- However, \( A \subset C \) does not imply that \( A \in C \). For \( A \) to be an element of \( C \), \( A \) itself must be one of the elements listed in set \( C \).
- For example, if \( C = \{1, 2, 3, 4, 5, 6\} \) and \( A = \{2, 4\} \), then \( A \subset C \) is true, but \( A \in C \) is false because \( A \) is not a single element in \( C \).

5. **Conclusion**:
- Therefore, the statement "If \( A \subset B \) and \( B \subset C \), then \( A \in C \)" is **false**.