Exercise · ZW-BIJECTIVE-FUNCTION-002 · ★☆☆ Basic
If \(f\colon A\to B\) is bijective, how many elements of \(A\) have image equal to a given \(b\in B\)?
Choose one option.
Show answer
- A. Exactly one
Explanation
Injectivity gives at most one preimage and surjectivity gives at least one, so each \(b\in B\) has exactly one preimage. The other three options describe only one of the two conditions, or neither.
Review the concept: Bijective function
Function \(f\colon A\to B\) that is both injective and surjective, so each element of \(B\) corresponds to exactly one element of \(A\).
\[\forall b\in B\;\exists!\,a\in A\,(f(a)=b)\]
More practice problems: bijective function
- ZW-BIJECTIVE-FUNCTION-001 Which two properties must fcolon A→ B satisfy to be bijective?
- ZW-BIJECTIVE-FUNCTION-003 What does it mean for fcolon A→ B to be injective?
- ZW-BIJECTIVE-FUNCTION-004 Which is the definition of surjectivity for fcolon A→ B?
- ZW-BIJECTIVE-FUNCTION-005 Let fcolon\1,2,3\→\a,b,c\ be injective. What can be concluded?
- ZW-BIJECTIVE-FUNCTION-006 Is fcolonℝ→ℝ, f(x)=x^3-x bijective?
- ZW-BIJECTIVE-FUNCTION-007 Which of these functions from ℤ to ℤ is bijective?
- ZW-BIJECTIVE-FUNCTION-008 If fcolon A→ B is bijective and |A|=5, what is |B|?
- ZW-BIJECTIVE-FUNCTION-009 Let fcolonℝ→ℝ, f(x)=x^3+x. Which statement is correct?