Exercise · ZW-SURJECTIVE-FUNCTION-009 · ★★★ Advanced
If \(f\colon A\to B\) and \(g\colon B\to C\) are surjective, is \(g\circ f\)?
Choose one option.
Show answer
- A. Yes; the composition of surjective functions is surjective
Explanation
Given \(c\in C\), the surjectivity of \(g\) gives \(b\in B\) with \(g(b)=c\), and that of \(f\) gives \(a\in A\) with \(f(a)=b\). Then \((g\circ f)(a)=c\). None of the other conditions is needed.
Review the concept: Surjective function
Function \(f\colon A\to B\) whose image is the whole codomain \(B\).
\[\forall b\in B\,\exists a\in A\,(f(a)=b)\]
More practice problems: surjective function
- ZW-SURJECTIVE-FUNCTION-001 What does it mean for fcolon A→ B to be surjective?
- ZW-SURJECTIVE-FUNCTION-002 Which of these functions from ℝ to ℝ is surjective?
- ZW-SURJECTIVE-FUNCTION-003 What is the logical form of the definition of a surjective function?
- ZW-SURJECTIVE-FUNCTION-004 If fcolon A→ B is not surjective, what holds?
- ZW-SURJECTIVE-FUNCTION-005 Let fcolon\1,2,3\→\a,b\. Can f be surjective?
- ZW-SURJECTIVE-FUNCTION-006 What is the name of the set f(A) of all values taken by f?
- ZW-SURJECTIVE-FUNCTION-007 Let fcolonℤ→ℤ, f(n)=2n. What is the image f(ℤ)?
- ZW-SURJECTIVE-FUNCTION-008 Let A=\1,2,3,4\ and B=\a,b\. How many surjective functions fcolon A→ B are there?