Problem 1 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(F_{1}\) be the set of all parallelograms, \(F_{2}\) the set of rectangles, \(F_{3}\) the set of rhombuses, \(F_{4}\) the set of squares and \(F_{5}\) the set of trapeziums in a plane then \(F_{1}\) is equal to (A) \(F_{2} \cap F_{3}\) (B) \(F_{2} \cup F_{3} \cup F_{4} \cup F_{1}\) (C) \(F_{3} \cap F_{4}\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Understand the Set Definitions

We need to understand the definitions of the sets:- \( F_{1} \): Parallelograms\- \( F_{2} \): Rectangles\- \( F_{3} \): Rhombuses\- \( F_{4} \): Squares\- \( F_{5} \): Trapeziums\Parallelograms are quadrilaterals with opposite sides parallel. Rectangles and rhombuses are specific types of parallelograms: rectangles have right angles, and rhombuses have equal sides. Squares are both rectangles and rhombuses. Trapeziums are quadrilaterals with exactly one pair of parallel sides.

Step 2: Examine the Options

We are given options for what \( F_{1} \), the set of all parallelograms, could equal in terms of intersections and unions of the other sets. We need to assess each option to determine if it is equivalent to \( F_{1} \).- **(A)**: \( F_{2} \cap F_{3} \) – This is the set of all squares since squares have both the properties of rectangles and rhombuses.- **(B)**: \( F_{2} \cup F_{3} \cup F_{4} \cup F_{1} \) – This is all rectangles, rhombuses, squares, and parallelograms, essentially covering all parallelograms and more.- **(C)**: \( F_{3} \cap F_{4} \) – This is only squares, as a square is the only shape that satisfies being both a rhombus and a square.- **(D)**: None of these.

Step 3: Analyze Each Option Against Parallelograms

We begin by checking if each option could represent all parallelograms:- **(A)**: \( F_{2} \cap F_{3} \) includes only squares, not all parallelograms.- **(B)**: While it includes \( F_{1} \), it also includes everything else, making it not equal to just all parallelograms.- **(C)**: Only includes squares.- **(D)** appears to suggest that none of the provided answers are encapsulating all and only parallelograms, considering the definitions and coverage in the other sets.

Step 4: Conclusion

The correct answer would have to be the set option that equalizes to only \( F_{1} \), but considering other options include either only parts of \( F_{1} \) or more than \( F_{1} \) itself without equating exactly to it, (D) None of These is the correct choice.

Key Concepts

Rectangles

Rectangles are a special type of parallelogram. They are quadrilaterals with opposite sides that are both parallel and equal in length. However, what makes rectangles unique is that all their interior angles are right angles – that means each angle is 90 degrees.
In the context of the problem, rectangles can be considered a subset of parallelograms. This is because they fulfill all the properties of a parallelogram, with the additional condition of having right angles. As such, every rectangle is a parallelogram, but not every parallelogram is a rectangle. The fact that rectangles form intersections with other sets, like rhombuses to form squares, demonstrates their unique role in understanding quadrilaterals.
Here are some key points about rectangles:

• All angles are 90 degrees.
• Opposite sides are parallel and equal.
• Diagonals bisect each other and are equal in length.

Understanding these properties sheds light on why rectangles intersect with rhombuses only at squares.

Rhombuses

Rhombuses are another type of parallelogram characterized by having all sides of equal length. Despite lacking the requirement of having right angles, like rectangles, rhombuses still maintain the essential property of opposite sides being parallel.
In the context of the exercise, it is helpful to consider rhombuses due to their ability to form intersections with rectangles to create squares. Notably, while all rectangles are not rhombuses, the opposite is also true, but squares find their place as an intersection of both.
Key characteristics of rhombuses include:

• All sides are equal in length.
• Opposite sides are parallel.
• The diagonals bisect each other at right angles.
• They have no right angles necessarily.

Comprehending these attributes explains their interaction within the set of parallelograms, but distinguishing them from rectangles requires special attention to side lengths and angles.

Squares

Squares are unique quadrilaterals that belong to the sets of both rectangles and rhombuses simultaneously. This means they possess all the properties of these shapes, making them an intriguing focal point in geometry.
A square exhibits features of a rectangle, with its equal right angles, and properties of a rhombus, with all equal sides. Thus, it serves as the intersection between the sets of rectangles and rhombuses, embodying the most specific form of a parallelogram.
The properties of squares include:

• All sides are equal in length.
• All angles are 90 degrees.
• Diagonals bisect each other and are equal in length.

In the original exercise, understanding squares helps unravel how intersections between the set of rectangles and the set of rhombuses yield this precise category of quadrilateral.

Trapeziums

Trapeziums stand apart from the other shapes discussed in the exercise. They are not parallelograms because they have exactly one pair of opposite sides that are parallel, unlike the two pairs in parallelograms.
Trapeziums do not fall into the same category as rectangles, rhombuses, or squares, but they offer a diverse look into quadrilaterals. This distinct trait makes them part of the problem's broader context without directly fitting into the set of parallelograms.
The main properties of trapeziums include:

• One pair of parallel sides, known as the bases.
• The other sides are non-parallel, known as the legs.

Understanding trapeziums is crucial, especially in problems requiring a comparison of different quadrilateral properties and when identifying the overall relationships within plane geometry.