Relations

Hardmathematics

The number of reflexive relations on a set with 3 elements is

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512

256

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Subject: mathematics
Chapter: sets, relations and functions
Topic: relations
Difficulty: Hard
Year: 2025

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Correct Answer:

Identify Matrix size: A set with $n = 3$ elements has $n^{2} = 9$ possible ordered pairs in the Cartesian product $A \times A$ .
Constraint of Reflexivity: For a relation to be reflexive, it must include the $n = 3$ diagonal pairs: ${(a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3})}$ .
Remaining Pairs: The non-diagonal pairs available for choice are $n^{2} - n = 9 - 3 = 6$ .
Calculate Combinations: Each of these 6 pairs can either be present or absent in the relation ( $2$ choices per pair).
Formula: Total reflexive relations $= 2^{(n^{2} - n)} = 2^{6} = 64$ .

This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of relations. It appeared in the 2025 exam.

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