Bijection#
A bijective function is a function that is both injective (one-to-one) and surjective (onto). It pairs every element in the domain with a unique element in the codomain, leaving no elements unmatched in either set, creating a direct one-to-one correspondence.
Key Characteristics
- Invertibility
A function has an inverse function (\(f^{-1}\)) if and only if it is a bijection.
- Cardinality Match
If a bijection exists between two finite sets, both sets have the exact same number of elements. This is also used to define equal sizes for infinite sets.
- Composition
The composition of two bijective functions is always another bijective function.
Source: GeeksforGeeks#
How to Prove#
To prove a function \(f: A → B\) is a bijection, you must prove it has two properties: it is subjective, and subjective.
Prove Injectivity.
Prove Surjectivity.