Prove the following :
(i) `AsubBhArrB^(c)subA^(c)`
(ii) `BsubArArrAuuB=A`.

Answer

To prove the statements given in the question, we will use the properties of sets and Venn diagrams.

### Proof of (i): If \( A \subseteq B \) then \( B^c \subseteq A^c \)

1. **Understanding the Definitions**:
- \( A \subseteq B \) means every element of set \( A \) is also an element of set \( B \).
- The complement of a set \( B \), denoted as \( B^c \), includes all elements in the universal set \( U \) that are not in \( B \).
- The complement of set \( A \), denoted as \( A^c \), includes all elements in the universal set \( U \) that are not in \( A \).

**Hint**: Recall the definitions of subset and complement.

2. **Visualizing with Venn Diagrams**:
- Draw a Venn diagram where circle \( A \) is completely inside circle \( B \). This represents \( A \subseteq B \).
- The area outside circle \( B \) represents \( B^c \), and the area outside circle \( A \) represents \( A^c \).

**Hint**: Sketch the Venn diagram to visualize the relationships.

3. **Analyzing the Areas**:
- Since \( A \) is entirely within \( B \), all elements not in \( B \) (i.e., in \( B^c \)) cannot be in \( A \). Hence, if an element is in \( B^c \), it must also be in \( A^c \).
- Therefore, all elements of \( B^c \) are also elements of \( A^c \).

**Hint**: Think about the elements that are excluded from both sets.

4. **Conclusion**:
- We conclude that \( B^c \subseteq A^c \) if \( A \subseteq B \).

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### Proof of (ii): If \( B \subseteq A \) then \( A \cup B = A \)

1. **Understanding the Definitions**:
- \( B \subseteq A \) means every element of set \( B \) is also an element of set \( A \).
- The union of sets \( A \) and \( B \), denoted as \( A \cup B \), includes all elements that are in \( A \), in \( B \), or in both.

**Hint**: Remember the definition of union.

2. **Visualizing with Venn Diagrams**:
- Draw a Venn diagram where circle \( B \) is completely inside circle \( A \). This represents \( B \subseteq A \).
- The union \( A \cup B \) will include all elements in circle \( A \) and circle \( B \).

**Hint**: Use the Venn diagram to see how the sets overlap.

3. **Analyzing the Areas**:
- Since all elements of \( B \) are already included in \( A \), when we take the union \( A \cup B \), we are essentially just collecting the elements of \( A \).
- Therefore, \( A \cup B \) does not add any new elements beyond those already in \( A \).

**Hint**: Consider what happens to the elements when you combine the two sets.

4. **Conclusion**:
- We conclude that \( A \cup B = A \) if \( B \subseteq A \).

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