Types of Functions

One-one (injective) function

A function $f : X \to Y$ is one-one (or injective) if distinct inputs always have distinct images. Precisely, for every $x_1, x_2 \in X$,

Equivalently, $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$: no two different elements of $X$ can share the same image. A function that is not one-one is called many-one. …

Onto (surjective) function

A function $f : X \to Y$ is onto (or surjective) if every element of the codomain is hit: for every $y \in Y$ there exists at least one $x \in X$ such that $f(x) = y$.

Equivalently, the range of $f$ equals the whole set $Y$ — no element of $Y$ is left without a pre-image. …

Definition 7: Bijective Function (One‑One and Onto)

A function $f : X \to Y$ is said to be one‑one and onto (or bijective) if it satisfies both conditions:

1. One‑one (injective): For every $x_1, x_2 \in X$,

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

   (Different inputs map to different outputs.)

2. Onto (surjective): For every $y \in Y$, there exists some $x \in X$ such that $f(x) = y$.

   (Every element of $Y$ is the image of at least one element of $X$.)

In short:

$f$ is bijective $\iff$ $f$ is both injective and surjective.

Intuition

Think of a bijection as a perfect pairing between the sets $X$ and $Y$:

• No two different $x$'s share the same $y$ (one‑one).
• Every $y$ gets matched to some $x$ (onto).

So a bijection is like a one‑to‑one correspondence — each element of $X$ is paired with exactly one element of $Y$, and vice versa.

Tiny Concrete Example

Let $X = \{1, 2, 3\}$ and $Y = \{a, b, c\}$. …

Fig 1.2 is a set of four mapping diagrams that visually define the core types of functions: one-one (injective), many-one, onto (surjective), and bijective. Each diagram shows a left oval labelled $X_1 = \{1,2,3,4\}$ (the domain) and a right oval (the codomain), with indigo arrows connecting elements.

In diagram (i), the function $f_1$ sends $1 \to a$, $2 \to b$, $3 \to d$, $4 \to c$ into $X_2 = \{a,b,c,d,e,f\}$. Every element of $X_1$ gets a distinct image — no two arrows land on the same element in $X_2$. This is a one-one (injective) map. Notice that $e$ and $f$ in the codomain have no incoming arrows; they are not the image of any element of $X_1$. So $f_1$ is not onto.

Diagram (ii) shows $f_2$: both $1$ and $2$ map to $b$, while $3 \to d$ and $4 \to f$. Because two different domain elements share the same image, this is a many-one function. It is also not onto, since $a$, $c$, and $e$ are unused.

In diagram (iii), $f_3$ maps into $X_3 = \{a,b,c\}$. Here $1$ and $2$ both go to $a$, $3 \to b$, $4 \to c$. Every element of $X_3$ receives at least one arrow — the function is onto (surjective). But because $1$ and $2$ share the same image, it is many-one, not one-one.

Diagram (iv) shows $f_4$ into $X_4 = \{a,b,c,d\}$ with crossing arrows: $1 \to b$, $2 \to a$, $3 \to d$, $4 \to c$. Every domain element has a distinct image (one-one), and every codomain element is hit (onto). This is a bijection — both one-one and onto.

The textbook uses these visual examples to introduce three formal definitions. A function $f: X \to Y$ is:

• One-one (injective) if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2 \in X$. Equivalently, distinct elements of $X$ map to distinct elements of $Y$.
• Onto (surjective) if for every $y \in Y$, there exists some $x \in X$ such that $f(x) = y$. This means the range of $f$ equals the entire codomain $Y$.
• Bijective if it is both one-one and onto.