Before You Begin

Question #41105

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ z \in \mathbb{C} \mid z^2 - 2iz - 2 = 0 \text{ and } \bar{z}^2 - 2i\bar{z} - 2 = 0 \}$ . Let $R$ be a relation on $S$ defined by $z_1 R z_2 \iff z_1 + z_2 = 0$ . Which of the following statements is true?

Options

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

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Reflexivity in Empty Sets|Quadratic Equations with Complex Coefficients|Conjugate Root Theorem

Why This Question Is Tricky

Common Mistakes

This question deploys a dual-layered trap.

First, it exploits the misapplication of the Conjugate Root Theorem; students habitually assume complex roots occur in conjugate pairs, overlooking that this only holds for polynomials with strictly real coefficients.

By forcing complex coefficients, the set of elements satisfying both the equation and its conjugate counterpart evaluates to the empty set.

Second, it tests the edge-case definition of relations on an empty set.

Students who incorrectly populate the set with the roots will find the relation symmetric and transitive but not reflexive (Option B), or if they include the conjugates, symmetric but neither reflexive nor transitive (Option C).

Conversely, students who correctly identify the empty set often falsely believe an empty relation cannot be reflexive (Option D), failing to recognize that universal quantifiers over an empty domain are vacuously true, making it a valid equivalence relation.

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