Exercise · ZW-SURJECTIVE-FUNCTION-010 · ★★☆ Intermediate
Let \(f\colon\{1,2\}\to\{a,b,c\}\). Can \(f\) be surjective?
Choose one option.
Show answer
- D. No; its image has at most \(2\) elements and \(\{a,b,c\}\) has \(3\)
Explanation
The domain has only two inputs, so the image has at most two values, while the codomain has three. No function can cover it. Being injective does not help either: it only requires the two values to be distinct.
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-008 Let A=\1,2,3,4\ and B=\a,b\. How many surjective functions fcolon A→ B are there?