The number of reflexive relations on a finite set A is 4096. The number of elements in A is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
Let A have n elements. Then \(A \times A\) has \(n^{2}\) pairs. A reflexive relation must contain the n pairs (a, a), and each of the remaining \(n^{2} - n\) pairs may be in or out:
number of reflexive relations \(= 2^{n^{2} - n}\)
Since \(4096 = 2^{12}\),
\(n^{2} - n = 12\)
\(n^{2} - n - 12 = 0\)
\((n - 4)(n + 3) = 0\)
As n cannot be negative, \(n = 4\).