Question

Let \( A = \{ 1, 2, 3, ..., n \} \), how many bijective functions \( f: A \to A \) can be defined?

• A. \( n \)
• B. \( n! \)
• C. \( \frac{1}{2} n \)
• D. \( (n-1)! \)

Hint:

The number of bijections from a set to itself is the same as the number of permutations of the set, which is \( n! \).

Answer 1

The Correct Option is B

For a function to be bijective, it must be both injective (one-to-one) and surjective (onto). This means that every element in the domain \( A \) must map to a unique and distinct element in the codomain \( A \). The number of ways to define a bijective function from a set of \( n \) elements to itself is given by the number of permutations of the set, which is \( n! \). Thus, the correct answer is \( n! \).