Exercise · ZW-SURJECTIVE-FUNCTION-004 · ★☆☆ Basic
If \(f\colon A\to B\) is not surjective, what holds?
Choose one option.
Show answer
- B. There exists \(b\in B\) that is not the image of any \(a\in A\)
Explanation
The negation of «every element of \(B\) is an image» is «some \(b\) has no preimage». The option about pairs with equal images describes failing injectivity, which does not follow from failing surjectivity: \(n\mapsto 2n\) from \(\mathbb{Z}\) to \(\mathbb{Z}\) is not surjective but is injective.
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-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?