f: A E

A function is considered a bijection, or a bijective function, when it satisfies two important properties: injectivity and surjectivity. A bijection is a special type of function between two sets where each element in the first set maps to a unique element in the second set, and every element in the second set has a preimage in the first set. Here are the formal definitions and properties of a bijection:

1. Injectivity (One-to-One): A function  is injective (or one-to-one) if, for every pair of distinct elements x₁​ and x₂​ in set $A$, their images under  $f$ in set  $B$ are also distinct. In other words, if f(x₁)=f(x_2​), then x_1​=x₂​.

2. Surjectivity (Onto): A function   is surjective (or onto) if for every element $y$ in set $B$, there exists at least one element $x$ in set $A$ such that $f(x)=y$. In simpler terms, the function covers the entire set $B$, and there are no "holes" in the mapping.

3. Bijectivity: A function  is bijective if it is both injective and surjective. In other words, it is a one-to-one correspondence between the elements of set $A$ and set $B$. This means that each element in set $A$ maps to a unique element in set $B$, and every element in set $B$ has a unique preimage in set $A$.

Properties of a bijection:

a. A bijection is invertible: A bijection has an inverse function f⁻¹ that maps elements from set $B$ back to set $A$. The inverse function f⁻¹ undoes the mapping of $f$. So, if $f$ maps $x$ to $y$, then f⁻¹ maps $y$ back to $x$.

b. The cardinality of sets $A$ and $B$ is the same: If $f$ is a bijection between $A$ and $B$, then the number of elements in $A$ is equal to the number of elements in $B$.

c. A bijection has a unique inverse: The inverse function f⁻¹ of a bijection $f$ is unique.

d. Composition of bijections is a bijection: If  and are both bijections, then their composition  is also a bijection.

Bijections are important in mathematics and various fields because they establish a one-to-one correspondence between sets, allowing for efficient data manipulation, set equivalence, and other applications in fields like set theory, combinatorics, and computer science.