Let `F_(1)` be the set of parallelograms, `F_(2)` the set of rectangle , `F_(3)` the set of rhombuses, `F_(4)` the set of squares and `F_(5)` the set of trapeziums in a plane. Then, `F_(1)` may be equal to

Options

A. `F_(2) nn F_(3)`
B. `F_(3) nn F_(4)`
C. `F_(2) uu F_(5)`
D. `F_(2) uu F_(3) uu F_(4) uu F_(1)`

Explanation

To solve the problem, we need to analyze the relationships between the different sets of quadrilaterals defined in the question. Let's break it down step by step.

### Step 1: Identify the Sets
We have the following sets:
- \( F_1 \): Set of parallelograms
- \( F_2 \): Set of rectangles
- \( F_3 \): Set of rhombuses
- \( F_4 \): Set of squares
- \( F_5 \): Set of trapeziums

### Step 2: Understand the Properties of Each Set
1. **Parallelogram**: A quadrilateral with opposite sides parallel.
2. **Rectangle**: A parallelogram with all angles equal to 90 degrees.
3. **Rhombus**: A parallelogram with all sides of equal length.
4. **Square**: A quadrilateral that is both a rectangle and a rhombus (all sides equal and all angles 90 degrees).
5. **Trapezium**: A quadrilateral with at least one pair of parallel sides, but not necessarily both pairs.

### Step 3: Determine Which Sets are Subsets of \( F_1 \)
- **Rectangles** (\( F_2 \)): All rectangles are parallelograms. Thus, \( F_2 \subseteq F_1 \).
- **Rhombuses** (\( F_3 \)): All rhombuses are parallelograms. Thus, \( F_3 \subseteq F_1 \).
- **Squares** (\( F_4 \)): All squares are parallelograms (since they are both rectangles and rhombuses). Thus, \( F_4 \subseteq F_1 \).
- **Trapeziums** (\( F_5 \)): Not all trapeziums are parallelograms. Thus, \( F_5 \not\subseteq F_1 \).

### Step 4: Formulate the Union of Relevant Sets
Since \( F_2 \), \( F_3 \), and \( F_4 \) are all subsets of \( F_1 \), we can express \( F_1 \) in terms of these sets:
\[ F_1 = F_2 \cup F_3 \cup F_4 \cup F_1 \]
This means that \( F_1 \) includes all rectangles, rhombuses, and squares, along with all parallelograms.

### Step 5: Conclusion
Thus, \( F_1 \) may be equal to the union of the sets of rectangles, rhombuses, and squares, along with itself. Therefore, the correct answer is:
\[ F_1 = F_2 \cup F_3 \cup F_4 \cup F_1 \]

### Final Answer
The correct option is:
- \( F_1 = F_2 \cup F_3 \cup F_4 \cup F_1 \)