Exercise · ZW-INJECTIVE-FUNCTION-009 · ★★☆ Intermediate
Is every injective function also surjective?
Choose one option.
Show answer
- D. No; \(n\mapsto n+1\) from \(\mathbb{N}\) to \(\mathbb{N}\) is injective and not surjective
Explanation
With \(\mathbb{N}=\{1,2,\dots\}\), \(n+1=n'+1\) implies \(n=n'\), so it is injective; but \(1\) is not \(n+1\) for any \(n\in\mathbb{N}\). The other options are false: a finite domain is not enough (for example \(\{1\}\to\{1,2\}\)), and injective functions from \(\mathbb{N}\) to \(\mathbb{N}\) exist, such as the identity.
Review the concept: Injective function
Function \(f\colon A\to B\) in which distinct elements of \(A\) have distinct images in \(B\).
\[\forall a_1,a_2\in A\,\bigl(f(a_1)=f(a_2)\Rightarrow a_1=a_2\bigr)\]
More practice problems: injective function
- ZW-INJECTIVE-FUNCTION-001 What does it mean for fcolon A→ B to be injective?
- ZW-INJECTIVE-FUNCTION-002 Which of these functions from ℝ to ℝ is injective?
- ZW-INJECTIVE-FUNCTION-003 What is the logical form of the definition of an injective function?
- ZW-INJECTIVE-FUNCTION-004 If fcolon A→ B is not injective, what holds?
- ZW-INJECTIVE-FUNCTION-005 Can there be an injective function fcolon\1,2,3\→\a,b\?
- ZW-INJECTIVE-FUNCTION-006 If f is injective and a_1ne a_2, what happens to f(a_1) and f(a_2)?
- ZW-INJECTIVE-FUNCTION-007 Let fcolonℤ→ℤ, f(n)=2n. Is f injective?
- ZW-INJECTIVE-FUNCTION-008 If fcolon A→ B and gcolon B→ C are injective, is g∘ f?