Exercise · ZW-INJECTIVE-FUNCTION-005 · ★★☆ Intermediate
Can there be an injective function \(f\colon\{1,2,3\}\to\{a,b\}\)?
Choose one option.
Show answer
- D. No, because the domain has more elements than the codomain
Explanation
With three inputs and only two possible outputs, two inputs must share an image by the pigeonhole principle. Every assignment repeats a value, so no such function is injective.
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-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?
- ZW-INJECTIVE-FUNCTION-009 Is every injective function also surjective?