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Question #41141

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . A binary operation $*$ on the power set $\mathcal{P}(S)$ is defined as $A * B = (A \setminus B) \cup (B \setminus A)$ . Let $\mathcal{I}$ be the family of all proper subsets of $S$ that do not contain any two consecutive integers. If $\mathcal{C}$ is the closure of $\mathcal{I}$ under the operation $*$ , find the number of elements in the set $\mathcal{V} = \{ A \in \mathcal{C} \setminus \{S\} \mid A \notin \mathcal{I} \}$ .

Options

A.
367
B.
422
C.
368
D.
423

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

Binary Operation Closure|Combinatorial Counting (P&C Integration)|Complementary Counting

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge abstract algebraic structures (Group Theory/Binary Closure) with advanced combinatorial counting.

The primary trap is the 'Lethal Collision' of attempting to manually construct the closure of $\mathcal{I}$ under symmetric difference, which leads to a combinatorial explosion.

Mathematically mature students will recognize that because $\mathcal{I}$ contains all singletons, it acts as a generating set for the entire Boolean group $(\mathcal{P}(S), \Delta)$ , making the closure trivially $\mathcal{P}(S)$ .

The secondary trap lies in the complementary counting: students must correctly identify the sequence of subsets with no consecutive integers (Fibonacci sequence, $F_{n+2}$ ) and carefully account for the exclusion of the universal set $S$ to arrive at the precise trap constant 367.

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