Bijection
A bijection (or bijective function) is a function that is both injective (one-to-one) and surjective (onto), meaning every element of the domain maps to a unique element of the codomain, and every element of the codomain is mapped to by exactly one element of the domain. Bijections establish a perfect one-to-one correspondence between two sets and are fundamental in defining cardinality, invertible functions, and isomorphisms across mathematics.
Key Formula
f is bijective iff for every y in B, there exists exactly one x in A such that f(x) = y
LaTeX: f \text{ is bijective} \iff \forall y \in B, \exists! x \in A : f(x) = y
| Symbol | Meaning | Unit |
|---|---|---|
| f | The function f: A → B | mapping |
| A | Domain of f | set |
| B | Codomain of f | set |
| x | Input element from A | element |
| y | Output element in B | element |
Worked Example
Problem
Prove that f: ℝ → ℝ defined by f(x) = 3x − 7 is a bijection.
Solution
Step 1: Prove injective (one-to-one). — Assume f(x₁) = f(x₂). — Then 3x₁ − 7 = 3x₂ − 7. — 3x₁ = 3x₂, so x₁ = x₂. ✓ (Injective) Step 2: Prove surjective (onto). — Let y ∈ ℝ be arbitrary. We need x ∈ ℝ with f(x) = y. — Set x = (y + 7)/3. Then f(x) = 3·(y+7)/3 − 7 = y + 7 − 7 = y. ✓ — Such an x always exists in ℝ. (Surjective) Since f is both injective and surjective, f is bijective.
Answer
f(x) = 3x − 7 is a bijection from ℝ to ℝ, with inverse f⁻¹(y) = (y + 7)/3.
Comparison of Injective, Surjective, and Bijective Functions
| Property | Injective | Surjective | Bijective |
|---|---|---|---|
| Alternative name | One-to-one | Onto | One-to-one correspondence |
| Condition | f(x₁)=f(x₂) ⟹ x₁=x₂ | Range = Codomain | Both conditions hold |
| Inverse exists? | Left inverse only | Right inverse only | Full inverse f⁻¹ exists |
| Example | f(x)=e^x (ℝ→ℝ) | f(x)=x³ (ℝ→ℝ) | f(x)=2x+1 (ℝ→ℝ) |
| Sets A and B size | |A| ≤ |B| | |A| ≥ |B| | |A| = |B| |
Interactive Tools
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Related Terms
Function (mathematics)
A function is a relation that assigns to each element of a set (called the domain) exactly one element of another set (called the codomain). Denoted f: A → B, it is a rule that maps every input to a unique output without ambiguity. Functions are among the most central concepts in mathematics and appear in calculus, linear algebra, programming, and virtually every applied science.
Domain (function)
The domain of a function is the complete set of all possible input values (x-values) for which the function is defined and produces a valid output. For a function f: A → B, the domain is the set A. Identifying the domain is a critical first step in analysing any function, as inputs outside the domain can lead to undefined expressions such as division by zero or square roots of negative numbers.
Permutation
A permutation is an arrangement of all or part of a set of objects in a specific order, where the order of selection matters. The number of permutations of r objects chosen from n distinct objects is denoted P(n, r) or nPr. Permutations are central to combinatorics, probability theory, and appear in computing contexts such as sorting algorithms and cryptography.
From French "bijection", coined by Nicolas Bourbaki in their landmark series "Éléments de mathématique" (1939). "Bi-" is from Latin "bis" (twice/both) + "jectio" from "jacere" (to throw), indicating mapping in both directions.