Exercise · ZW-SURJECTIVE-FUNCTION-008 · ★★★ Advanced
Let \(A=\{1,2,3,4\}\) and \(B=\{a,b\}\). How many surjective functions \(f\colon A\to B\) are there?
Choose one option.
Show answer
- B. 14
Explanation
There are \(2^4=16\) functions in total. Subtract the two constant functions, which miss one element of \(B\): \(16-2=14\). All the others take both values.
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-009 If fcolon A→ B and gcolon B→ C are surjective, is g∘ f?