Injective, Surjective, and Bijective Functions

In mathematics, a function describes how elements of one set are assigned to elements of another. Three important properties capture how completely and uniquely a function connects its domain to its codomain: injective, surjective, and bijective. These properties tell us whether different inputs can share the same output and whether every possible output is used.

Injective (One-to-One)

A function is injective if different inputs always produce different outputs. No two distinct elements in the domain are allowed to map to the same result in the codomain.

Formally, a function f : A → B is injective if:

f(a₁) = f(a₂) ⇒ a₁ = a₂

Equivalently, if a₁ ≠ a₂, then f(a₁) ≠ f(a₂).

Example: f(x) = ex from ℝ → ℝ is injective. Different inputs produce different outputs, but not every real number is reached, so it is not surjective.

Surjective (Onto)

A function is surjective if every element of the codomain is reached by the function — nothing is left out. That is, for every element b ∈ B, there exists at least one element a ∈ A such that f(a) = b.

Every element of the codomain is covered.

Example: f(x) = x² from ℝ → ℝ_≥0 is surjective. Every non-negative real number has a real square root, but the function is not injective because different inputs can produce the same output (e.g., 2 and −2).

Bijective (One-to-One and Onto)

A function is bijective if it is both injective and surjective. This means:

• Different inputs give different outputs (injective), and
• Every element of the codomain is used (surjective).

A bijection creates a perfect pairing between domain and codomain. Because of this, bijective functions are reversible: each output corresponds to exactly one input, so an inverse function exists.

Example: f(x) = x³ from ℝ → ℝ is bijective. It is injective (different inputs give different outputs) and surjective (every real number is reached).

Summary Table

Type	Description	Inverse Exists?
Injective	Different inputs give different outputs	No (not necessarily)
Surjective	Every element of the codomain is reached	No (not necessarily)
Bijective	Both injective and surjective	Yes

Why These Ideas Matter

These properties help us understand how precisely a function connects one set to another. Injective functions tell us that outputs are never duplicated. Surjective functions guarantee that the codomain is fully covered. Bijective functions achieve both conditions, giving a perfect one-to-one correspondence. Because bijections can be reversed, they are especially important in algebra, analysis, and many other areas of mathematics.