Given, A = {Quadrilaterals}, B = {Rectangles}, C = {Squares}, D = {Rhombuses}. State, giving reasons, whether the following are true or false.
(i) `B sub C`, (ii) `D sub B`, (iii) `C sube B sube A`, (iv) `D sub A`, (v) `B supe C`, (iv) `A supe B supe C`

Answer

To determine the truth of each statement regarding the sets A, B, C, and D, we need to analyze the relationships between these sets based on their definitions.

Given:
- A = {Quadrilaterals}
- B = {Rectangles}
- C = {Squares}
- D = {Rhombuses}

Now, let's evaluate each statement one by one.

### Step-by-Step Solution:

1. **Statement (i): B ⊆ C**
- This statement claims that all rectangles are squares.
- **Reasoning**: A rectangle is defined as a quadrilateral with opposite sides equal and all angles equal to 90 degrees. A square is a special type of rectangle where all four sides are equal. Therefore, not all rectangles are squares.
- **Conclusion**: This statement is **False**.

2. **Statement (ii): D ⊆ B**
- This statement claims that all rhombuses are rectangles.
- **Reasoning**: A rhombus is a quadrilateral with all sides equal, but it does not necessarily have right angles. Therefore, not all rhombuses are rectangles.
- **Conclusion**: This statement is **False**.

3. **Statement (iii): C ⊆ B ⊆ A**
- This statement claims that all squares are rectangles and all rectangles are quadrilaterals.
- **Reasoning**: A square is indeed a rectangle (since it meets the criteria of a rectangle) and all rectangles are quadrilaterals. Thus, this statement holds true.
- **Conclusion**: This statement is **True**.

4. **Statement (iv): D ⊆ A**
- This statement claims that all rhombuses are quadrilaterals.
- **Reasoning**: A rhombus is defined as a quadrilateral with all sides equal. Therefore, it is indeed a type of quadrilateral.
- **Conclusion**: This statement is **True**.

5. **Statement (v): B ⊇ C**
- This statement claims that rectangles are a superset of squares.
- **Reasoning**: While all squares are rectangles, not all rectangles are squares. Therefore, rectangles cannot be considered a superset of squares.
- **Conclusion**: This statement is **False**.

6. **Statement (vi): A ⊇ B ⊇ C**
- This statement claims that quadrilaterals are a superset of rectangles and rectangles are a superset of squares.
- **Reasoning**: While it is true that all rectangles are quadrilaterals and all squares are rectangles, the statement implies a direct superset relationship which is misleading. Quadrilaterals include many shapes that are not rectangles.
- **Conclusion**: This statement is **False**.

### Summary of Results:
- (i) False
- (ii) False
- (iii) True
- (iv) True
- (v) False
- (vi) False