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bijection
This term describes a perfect one-to-one correspondence between two collections. It implies a state of total symmetry where no element is left unpaired and no element is shared by multiple partners, creating a mathematical bridge that allows for a complete and reversible mapping.
In practical application, the term is used almost exclusively within formal mathematics, specifically set theory and algebra. It carries a connotation of precision and exclusivity, distinguishing it from general functions or injections where such a strict, bidirectional pairing is not required.
Meanings
Examples
A bijection ensures that every element in the domain has a unique partner in the codomain.
Is this mapping a bijection or just an injection?
The professor explained that a bijection is both injective and surjective.
I need to find a bijection to prove these two sets have the same cardinality.
The function is a bijection because it is strictly increasing and covers the entire range.
If we can establish a bijection, the two structures are essentially identical.
Wait, if it is not a bijection, then the inverse function does not exist.
The proof relies on the existence of a bijection between the two infinite sets.