Exercise · ZW-SURJECTIVE-FUNCTION-003 · ★☆☆ Basic
What is the logical form of the definition of a surjective function?
Choose one option.
Show answer
- C. \(\forall b\in B\,\exists a\in A\,(f(a)=b)\)
Explanation
Every \(b\) in the codomain needs some \(a\) in the domain that reaches it; the order of the quantifiers matters. The negation of surjectivity is the option stating that some \(b\) is reached by no \(a\), and the one comparing images describes injectivity. The option with a single \(a\) for all \(b\) swaps the quantifiers.
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-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?
- ZW-SURJECTIVE-FUNCTION-009 If fcolon A→ B and gcolon B→ C are surjective, is g∘ f?