Exercise · ZW-BIJECTIVE-FUNCTION-007 · ★★☆ Intermediate
Which of these functions from \(\mathbb{Z}\) to \(\mathbb{Z}\) is bijective?
Choose one option.
Show answer
- D. \(f(n)=n+1\)
Explanation
The map \(n\mapsto n+1\) has inverse \(m\mapsto m-1\), so it is bijective. The other three fail: \(2n\) never hits odd integers, while \(n^2\) and \(|n|\) send \(n\) and \(-n\) to the same value.
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-008 If fcolon A→ B is bijective and |A|=5, what is |B|?
- ZW-BIJECTIVE-FUNCTION-009 Let fcolonℝ→ℝ, f(x)=x^3+x. Which statement is correct?