Exercise · ZW-SURJECTIVE-FUNCTION-005 · ★★☆ Intermediate
Let \(f\colon\{1,2,3\}\to\{a,b\}\). Can \(f\) be surjective?
Choose one option.
Show answer
- A. Yes, for example with \(f(1)=a\), \(f(2)=b\), \(f(3)=b\)
Explanation
The three inputs can be distributed to cover both \(a\) and \(b\): one must map to \(a\) and another to \(b\). A larger domain does not prevent surjectivity; what prevents it is an image with fewer values than \(B\).
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-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?
- ZW-SURJECTIVE-FUNCTION-009 If fcolon A→ B and gcolon B→ C are surjective, is g∘ f?