Exercise · ZW-INJECTIVE-FUNCTION-006 · ★☆☆ Basic
If \(f\) is injective and \(a_1\ne a_2\), what happens to \(f(a_1)\) and \(f(a_2)\)?
Choose one option.
Show answer
- C. They are distinct
Explanation
It is the contrapositive of the definition: if the images matched, the points would have to be equal, contradicting \(a_1\ne a_2\). So the images are distinct.
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-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?