Exercise · ZW-INJECTIVE-FUNCTION-003 · ★☆☆ Basic
What is the logical form of the definition of an injective function?
Choose one option.
Show answer
- B. \(\forall a_1,a_2\in A\,(f(a_1)=f(a_2)\Rightarrow a_1=a_2)\)
Explanation
The first quantifier ranges over the whole domain, and the implication says that equal images force equal points. The option with \(\forall b\in B\) is surjectivity, not injectivity.
Review the concept: Injective function
Function \(f\colon A\to B\) in which distinct elements of \(A\) have distinct images in \(B\).
\[\forall a_1,a_2\in A\,\bigl(f(a_1)=f(a_2)\Rightarrow a_1=a_2\bigr)\]
More practice problems: injective function
- ZW-INJECTIVE-FUNCTION-001 What does it mean for fcolon A→ B to be injective?
- ZW-INJECTIVE-FUNCTION-002 Which of these functions from ℝ to ℝ is injective?
- ZW-INJECTIVE-FUNCTION-004 If fcolon A→ B is not injective, what holds?
- ZW-INJECTIVE-FUNCTION-005 Can there be an injective function fcolon\1,2,3\→\a,b\?
- ZW-INJECTIVE-FUNCTION-006 If f is injective and a_1ne a_2, what happens to f(a_1) and f(a_2)?
- ZW-INJECTIVE-FUNCTION-007 Let fcolonℤ→ℤ, f(n)=2n. Is f injective?
- ZW-INJECTIVE-FUNCTION-008 If fcolon A→ B and gcolon B→ C are injective, is g∘ f?
- ZW-INJECTIVE-FUNCTION-009 Is every injective function also surjective?