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Question #41107

Difficulty: 8/10 | 🟢 Free

The Question

Let $X = \{1, 2, 3, \dots, 20\}$ . For each $x \in X$ , define the set $S_x$ as the interval: $S_x = \left[ \left\lfloor \frac{x}{4} \right\rfloor, \left\lfloor \frac{x}{4} \right\rfloor + \log_2\left(1 + \sin^2\left(\frac{\pi x}{2}\right)\right) \right)$ where $\lfloor \cdot \rfloor$ denotes the greatest integer function. A relation $R$ is defined on $X$ such that $x R y$ if and only if $S_x \cap S_y \neq \emptyset$ . Which of the following statements is true?

Options

A.
$R$ is symmetric and transitive, but not reflexive
B.
$R$ is an equivalence relation
C.
$R$ is reflexive and symmetric, but not transitive
D.
$R$ is symmetric, but neither reflexive nor transitive

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Reflexivity in Empty Sets|Equivalence Class Partitioning|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the formal definition of reflexivity against the properties of empty sets generated by half-open intervals.

Students often see the floor function and immediately assume it creates a valid equivalence class partition.

The trap lies in the interval $[k, k)$ , which is the empty set $\emptyset$ .

Because $S_x$ is empty for even $x$ , the intersection $S_x \cap S_x$ is also empty, meaning the relation fails reflexivity.

It forces students to distinguish between a 'set defined by an interval formula' and a 'set containing elements', highlighting that reflexivity must hold for every element in the domain, without exception.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Reflexivity in Empty Sets
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free