What does bijection mean?
A bijection is a special kind of mathematical function that establishes an exact, unbreakable pairing between the elements of two sets. To qualify as a bijection, a function must be both injective — meaning no two different inputs produce the same output — and surjective, meaning every element of the target set is actually reached. The result is a perfect match: each element on one side corresponds to precisely one element on the other, and nothing is left unmatched. This property gives bijections enormous power in mathematics, because they are invertible — knowing the pairing in one direction lets you reverse it completely. Bijections are central to counting arguments, where showing a bijection between two collections proves they have the same size, and to the study of infinite sets, where Georg Cantor famously used them to compare different sizes of infinity. Outside pure mathematics, the term appears in computer science, logic, and cryptography wherever exact correspondences matter. It is formal, precise vocabulary, often paraphrased as a 'one-to-one correspondence' in less technical discussion.
nounA mathematical function between two sets that pairs every element of one set with exactly one element of the other, and vice versa — a mapping that is both injective (one-to-one) and surjective (onto).
- A function between two sets that is both injective (one-to-one) and surjective (onto), pairing each element of one set with exactly one element of the other.
"The function f(x) = 2x + 1 is a bijection from the integers to the odd numbers."
"There is a simple bijection between the natural numbers and their squares: just pair n with n²."
"To show the two groups are isomorphic, she first constructed a bijection between their underlying sets."
"Because a bijection has an inverse, it lets you translate back and forth between the two sets without ambiguity."
Standard plural, formed regularly by adding -s. Used when discussing multiple such mappings or classes of mappings.
"There are exactly six bijections between any two sets of three elements."
Cantor used bijections to prove there are exactly as many even numbers as whole numbers — and then shocked mathematicians by showing some infinities are bigger than others.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of bijection
The word 'bijection' was coined in twentieth-century mathematics from Latin roots: 'bi-' meaning 'two' or 'both', and 'jacere', meaning 'to throw'. It combines the prefix 'bi-' with '-jection', the same root seen in 'injection', 'surjection', and 'projection', which describe related types of mappings introduced around the same period by the French school of mathematics, particularly through the work of Nicolas Bourbaki's circle. The term entered English usage alongside these cognates as set theory became the standard foundation of mathematics.
Related word forms
How bijection is actually used
Strictly technical vocabulary, confined to mathematics and closely related fields such as theoretical computer science; it would sound out of place in general prose. In informal mathematical speech, people often say 'one-to-one correspondence' instead, though that phrase can be ambiguous since 'one-to-one' alone sometimes means merely injective. The adjective form 'bijective' is common in describing functions.
Easily confused with bijection
An injection maps distinct inputs to distinct outputs but need not cover the whole target set, whereas a bijection must also hit every element of that target set.
Bisection means dividing something into two parts (as in the bisection method for finding roots), while bijection is a one-to-one and onto mapping between sets.