Injective and surjective maps

Japanese school year: University year 1

What you learn

You will learn the formal definitions of injective, surjective, and bijective mappings, as well as the conditions required for inverse mappings to exist. These concepts are fundamental in modern algebra and invertible data transformations in computer science. Prior familiarity with functions and basic set operations will support your progress.

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Key points

This defines an injective (one-to-one) function $f$ . It states that if two outputs are equal ( $f(x_1) = f(x_2)$ ), their inputs $x_1$ and $x_2$ must be identical, meaning distinct inputs never share an output.

f(x_1) = f(x_2) \implies x_1 = x_2

This defines a surjective (onto) function $f$ . It means every element $y$ in the target set $Y$ has at least one source element $x$ in set $X$ satisfying $f(x) = y$ , leaving no target element uncovered.

\forall y \in Y, \; \exists x \in X \; (f(x) = y)

When a function $f$ is bijective (both injective and surjective), it has an inverse function $f^{-1}$ that reverses it. Applying one after the other returns the input unchanged via the identity map $\mathrm{id}$ .

f^{-1} \circ f = \mathrm{id}_X, \quad f \circ f^{-1} = \mathrm{id}_Y

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Practice 1Coming soon