Injective (1-1) and surjective (onto) functions

We introduce three fundamental properties that some functions have. These properties test your ability to work with quantifiers in a very fundamental way.

Injective functions

Definition (12.4 in the book): Let \(f:A\to B\) be a function. Then

  • \(f\) is called injective if, for all \(a,a'\) in \(A\), if \(a\not=a'\) then \(f(a)\not=f(a')\). (Such \(f\) are also called “one-to-one” functions).

Surjective functions

  • \(f\) is called surjective if, for all \(b\in B\), there exists \(a\in A\) such that \(f(a)=b\). (such \(f\) are also called “onto” functions.)

Note: whether a function is surjective depends on its codomain. It is always surjective onto its range.

Picture from page 229

Bijective functions

  • \(f\) is called bijective if it is both surjective and injective.