Injectivity, surjectivity, and bijectivity

Definition: Injective, surjective, and bijective functions

Suppose $f:X\to Y$ is a function. We say $f$ is

• Injective if for all $x\in X$, $f(x_1)=f(x_2)$ implies that $x_1=x_2$.
  • Contrapositive: any two distinct elements in the domain map to distinct elements in the codomain.
• Surjective if for each $y\in Y$, there exists $x\in X$ such that $f(x)=y$; that is, the image of $f$ is the whole codomain.
• Bijective if $f$ is both injective and surjective.

A function’s injectivity and surjectivity depends entirely on its domain and codomain.

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Review

• If $f(x)=x^2$ injective, surjective, or bijective? How can the domain and codomain be restricted to satisfy each definition?