Exercise · ZW-INJECTIVE-FUNCTION-001 · ★☆☆ Basic
What does it mean for \(f\colon A\to B\) to be injective?
Choose one option.
Show answer
- D. Distinct elements of \(A\) have distinct images
Explanation
Injective means the function never repeats values: two distinct points of the domain never share an image. The option about covering all of \(B\) describes surjectivity, and the other two are not equivalent to injectivity.
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-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?
- ZW-INJECTIVE-FUNCTION-009 Is every injective function also surjective?