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Question #41142

Difficulty: 8/10 | 🟢 Free

The Question

Let a binary operation $*$ be defined on a subset $S \subseteq \mathbb{R}$ as $a * b = \left[ \text{sgn}(a) \cos^2\left(\frac{\pi b}{2}\right) \right]$ where $[\cdot]$ denotes the greatest integer function and $\text{sgn}(\cdot)$ denotes the signum function. For which of the following sets $S$ is $*$ a closed binary operation?

Options

A.
$(-1, 1)$
B.
$[-1, 1)$
C.
$(-1, 1]$
D.
$[-1, 1]$

Check Your Answer

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

Binary Operation Closure|Greatest Integer Function (GIF)|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the concept of closure in binary operations combined with the discontinuous nature of the Signum and Greatest Integer Functions.

The specific trap lies in assuming that because the operands $a$ and $b$ are strictly bounded within an open or half-open interval like $(-1, 1)$ , the result of the operation will also strictly remain within that interval.

However, the Signum function maps any positive fraction to exactly $1$ , and $\cos^2(0)$ is exactly $1$ , causing their product to hit the integer boundary.

The GIF then evaluates this exactly to $1$ or $-1$ , breaking closure for any set that does not explicitly include these boundary endpoints.

Students failing to check the exact integer transitions at $b=0$ will incorrectly select the open or half-open intervals.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Binary Operation Closure
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free