Exercise · ZW-INJECTIVE-FUNCTION-008 · ★★★ Advanced
If \(f\colon A\to B\) and \(g\colon B\to C\) are injective, is \(g\circ f\)?
Choose one option.
Show answer
- A. Yes; the composition of injective functions is injective
Explanation
If \((g\circ f)(a_1)=(g\circ f)(a_2)\), the injectivity of \(g\) gives \(f(a_1)=f(a_2)\), and that of \(f\) gives \(a_1=a_2\). None of the other conditions is needed.
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-009 Is every injective function also surjective?