Exercise · ZW-SURJECTIVE-FUNCTION-007 · ★★☆ Intermediate
Let \(f\colon\mathbb{Z}\to\mathbb{Z}\), \(f(n)=2n\). What is the image \(f(\mathbb{Z})\)?
Choose one option.
Show answer
- C. The even numbers
Explanation
Each value \(2n\) is even, and every even number \(2m\) is obtained with \(n=m\). So the image is the even numbers, and \(f\) is not surjective because the odd numbers are never reached.
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-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?