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Question #41100

Difficulty: 8/10 | 🟢 Free

The Question

Let $X = \{1, 2, 3\}$ . A relation $R$ on $X$ is chosen at random from the set of all possible relations on $X$ . Let $E_1$ be the event that $R$ is the universal relation on $X$ and is also anti-symmetric. Let $E_2$ be the event that $R$ is an equivalence relation on $X$ . Which of the following statements is true regarding the events $E_1$ and $E_2$ ?

Options

A.
$E_1$ and $E_2$ are mutually exclusive and independent
B.
$E_1$ and $E_2$ are mutually exclusive but dependent
C.
$E_1$ and $E_2$ are independent but not mutually exclusive
D.
$E_1$ and $E_2$ are neither mutually exclusive nor independent

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

Empty and Universal Relation|Sample Space Construction|Mutually Exclusive vs Independent

Why This Question Is Tricky

Common Mistakes

This question tests the boundary conditions of probability axioms, specifically the intersection of mutually exclusive and independent events.

Students often rely on the heuristic that 'mutually exclusive events are dependent' because the occurrence of one prevents the other.

However, this heuristic fails when one of the events is an empty set (probability zero).

By constructing an event space based on an impossible relation (universal yet anti-symmetric), the question forces $P(E_1) = 0$ , making the events trivially both mutually exclusive and independent, trapping students who do not strictly apply the mathematical definitions $A \cap B = \emptyset$ and $P(A \cap B) = P(A)P(B)$ .

Quick Facts About This Question

Question Information

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