Bijection

Bijection in Set Theory

A bijection (or bijective function) is a function that is both injective and surjective — that is, a one-to-one correspondence between two sets.

Definition

Given two sets $A$ and $B$ , a function

f: A \to B

is a bijection if:

Injective (one-to-one):
$\forall a_1, a_2 \in A,\ f(a_1) = f(a_2) \implies a_1 = a_2$
Surjective (onto):
$\forall b \in B,\ \exists a \in A \text{ such that } f(a) = b$

If both conditions hold, each element of $A$ corresponds to exactly one unique element of $B$ , and vice versa.

Examples

Finite example:

A = \{1, 2, 3\}, \quad B = \{a, b, c\}

Define $f(1)=a,\ f(2)=b,\ f(3)=c$ .
✅ This is a bijection — each element of $A$ matches exactly one in $B$ .

Infinite example:

f: \mathbb{Z} \to \mathbb{Z}, \quad f(x) = x + 1

✅ This is also bijective, since it’s both injective and surjective.

Why It Matters

If there exists a bijection between two sets $A$ and $B$ , they have the same cardinality (the same “size”), even if they’re infinite.
For example:

\mathbb{N} \leftrightarrow \text{even numbers } 2\mathbb{N}

In Set Theory vs. Real Analysis

Context	Description
Set theory	Purely structural — about correspondence between sets. Used to define cardinality.
Real analysis	Adds extra properties (continuity, differentiability). Bijection helps define invertible functions like $f(x) = e^x$ and its inverse $\ln(x)$ .