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Question #41897

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7\}$ and $X$ be the power set of $S$ . A relation $R$ on $X$ is defined by $A \ R \ B$ if and only if $||A| - |B|| = (-1)^{|A \cup B|} (|A| - |B|)$ and the set $A \cup B$ contains no two consecutive integers. Find the number of ordered pairs $(A, B) \in X \times X$ such that $A \ R \ B$ .

Options

A.
325
B.
508
C.
142
D.
141

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

Symmetry vs. Antisymmetry|String Method (Adjacency)|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to deconstruct a parity-dependent modulus relation integrated with combinatorial adjacency constraints.

The primary trap lies in the relation $||A| - |B|| = (-1)^{|A \cup B|} (|A| - |B|)$ .

Students often misinterpret this as a symmetric equivalence relation (leading to 142 pairs where $|A|=|B|$ ) or ignore the modulus entirely to just calculate all disjoint/intersecting subsets (leading to 508 pairs).

The modulus sign dependency actually breaks symmetry, creating a conditional partial order that forces $|A| \ge |B|$ or $|A| \le |B|$ based on the parity of the union's size.

Successfully navigating this requires recognizing that the directional inequality perfectly halves the non-equal pairs, necessitating the use of central trinomial coefficients to account for the $|A|=|B|$ boundary cases.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Symmetry vs. Antisymmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free