Exercise · ZW-BIJECTIVE-FUNCTION-010 · ★★☆ Intermediate
How many bijections are there from a set \(A\) with \(n\) elements to itself?
Choose one option.
Show answer
- A. \(n!\)
Explanation
A bijection from \(A\) to itself is a permutation of its elements. Choosing the image of each element leaves \(n\), \(n-1\), \(\dots\), 1 options without repetition, giving \(n!\). \(2^n\) counts the subsets of \(A\), while \(n\) and \(n^2\) do not count permutations.
Review the concept: Bijective function
Function \(f\colon A\to B\) that is both injective and surjective, so each element of \(B\) corresponds to exactly one element of \(A\).
\[\forall b\in B\;\exists!\,a\in A\,(f(a)=b)\]
More practice problems: bijective function
- ZW-BIJECTIVE-FUNCTION-001 Which two properties must fcolon A→ B satisfy to be bijective?
- ZW-BIJECTIVE-FUNCTION-002 If fcolon A→ B is bijective, how many elements of A have image equal to a given b∈ B?
- ZW-BIJECTIVE-FUNCTION-003 What does it mean for fcolon A→ B to be injective?
- ZW-BIJECTIVE-FUNCTION-004 Which is the definition of surjectivity for fcolon A→ B?
- ZW-BIJECTIVE-FUNCTION-005 Let fcolon\1,2,3\→\a,b,c\ be injective. What can be concluded?
- ZW-BIJECTIVE-FUNCTION-006 Is fcolonℝ→ℝ, f(x)=x^3-x bijective?
- ZW-BIJECTIVE-FUNCTION-007 Which of these functions from ℤ to ℤ is bijective?
- ZW-BIJECTIVE-FUNCTION-008 If fcolon A→ B is bijective and |A|=5, what is |B|?