A,B and C are three set such that `A in B and B sub C`, then is it true that `A sub C` ? If not, give an example.

Answer

To determine whether the statement "If A is in B and B is a subset of C, then A is a subset of C" is true or not, we can analyze the definitions of set membership and subset relationships.

### Step-by-step Solution:

1. **Understanding the Given Information**:
- We have three sets: A, B, and C.
- The statement says that A is an element of B (denoted as \( A \in B \)).
- It also states that B is a subset of C (denoted as \( B \subseteq C \)).

2. **Definitions**:
- When we say \( A \in B \), it means that A is one of the elements in the set B.
- When we say \( B \subseteq C \), it means that every element of B is also an element of C.

3. **Analyzing the Relationship**:
- The statement we need to evaluate is whether \( A \subseteq C \) must be true given the above conditions.
- Since \( A \in B \), A itself is an element of B, not a subset. This means A can be a single element or a set itself, but it does not imply that all elements of A are contained in B.

4. **Counterexample**:
- Let's provide a specific example to show that the statement is not necessarily true.
- Let \( A = \{1\} \), \( B = \{\{1\}, 2, 3\} \), and \( C = \{1, 2, 3, 4\} \).
- Here, \( A \in B \) because the set A (which is \{1\}) is one of the elements in B.
- Also, \( B \subseteq C \) because all elements of B (\{1\}, 2, and 3) are present in C.
- However, \( A \subseteq C \) is not true because the set A contains the element 1, but as a set, it is not directly an element of C.

5. **Conclusion**:
- Therefore, it is not necessarily true that \( A \subseteq C \) when \( A \in B \) and \( B \subseteq C \).
- The example provided demonstrates that \( A \) can be an element of \( B \) while not being a subset of \( C \).