Bijection is a mathematical concept that is used to describe the relationship between two sets. It is a type of function that is both injective and surjective, meaning that each element in one set is paired with a unique element in the other set, and vice versa. In this article, we will explore the definition and meaning of bijection, its origin, and its associations in different contexts.
Definitions
A bijection is a function that maps each element of a set to a unique element of another set, such that every element in the second set is paired with exactly one element in the first set. In other words, a bijection is a one-to-one correspondence between two sets, where every element in one set has a corresponding element in the other set, and vice versa.
Origin
The term “bijection” comes from the French word “bijection”, which means “two-way function”. The concept of bijection has been used in mathematics for centuries, but the term itself was first introduced in the early 20th century by the French mathematician Émile Borel.
Meaning in different dictionaries
In the Oxford English Dictionary, bijection is defined as “a function that is both injective and surjective, that is, a function that maps each element of one set to a unique element of another set, and vice versa”. The Merriam-Webster Dictionary defines it as “a mathematical function that assigns to each member of one set exactly one member of another set”.
Associations
Bijection is a fundamental concept in mathematics, and it is used in a wide range of fields, including algebra, topology, and analysis. It is also used in computer science, particularly in the design and analysis of algorithms, as well as in cryptography and coding theory.
Synonyms
Some synonyms of bijection include one-to-one correspondence, bijective function, and invertible function.
Antonyms
The antonyms of bijection are functions that are not injective or not surjective. A function that is not injective is called a many-to-one function, while a function that is not surjective is called a partial function.
The same root words
The root word of bijection is “ject”, which means “to throw” or “to cast”. Other words that share this root include “eject”, “object”, and “project”.
Example Sentences
- The function f(x) = 2x is a bijection between the set of even numbers and the set of integers.
- In cryptography, a bijection is used to encrypt and decrypt messages.
- The concept of bijection is important in topology, where it is used to study the properties of continuous functions.
- A function that is not a bijection may still be useful in certain contexts, such as when dealing with partial information or incomplete data.