A, B and C are three non-empty sets. If `A sub B and B subC` then which the following is true?

Options

A. `B-A=C-B`
B. `AnnBnnC=B`
C. `AuuB=BnnC`
D. `AuuBuuC=A`

Answer

To solve the problem, we need to analyze the relationships between the sets A, B, and C given that \( A \subseteq B \) and \( B \subseteq C \). We will evaluate each option to determine which one is true.

### Step-by-Step Solution:

1. **Understanding the Subset Relationships**:
- We know that \( A \subseteq B \) means all elements of set A are also in set B.
- Similarly, \( B \subseteq C \) means all elements of set B are also in set C.
- Therefore, we can conclude that \( A \subseteq B \subseteq C \).

2. **Evaluating the Options**:
- We will analyze each option provided in the question.

3. **Option 1: \( B - A = C - B \)**:
- \( B - A \) represents the elements in B that are not in A.
- \( C - B \) represents the elements in C that are not in B.
- Since A is a subset of B and B is a subset of C, the elements of B that are not in A will not equal the elements of C that are not in B.
- Therefore, this option is **false**.

4. **Option 2: \( A \cap B \cap C = B \)**:
- \( A \cap B \cap C \) represents the elements common to all three sets.
- Since A is a subset of B, the intersection will actually be A: \( A \cap B \cap C = A \).
- Thus, this option is also **false**.

5. **Option 3: \( A \cup B = B \) and \( B \cap C = B \)**:
- \( A \cup B \) is the union of A and B, which will equal B since A is a subset of B.
- \( B \cap C \) is the intersection of B and C, which will also equal B since B is a subset of C.
- Therefore, this option is **true**.

6. **Option 4: \( A \cup B \cup C = A \)**:
- \( A \cup B \cup C \) represents the union of all three sets.
- Since C contains all elements of A and B, this union cannot equal A alone.
- Thus, this option is **false**.

### Conclusion:
The only true statement among the options is **Option 3**: \( A \cup B = B \) and \( B \cap C = B \).