A bijection is a function between two sets that is both injective (one-to-one) and surjective (onto). This means that every element in the target set has exactly one unique element in the domain that maps to it, and every element in the domain maps to a unique element in the target set. In simpler terms, a bijective function pairs each element of one set with exactly one element of another set, covering all elements in both sets without any repetition or omissions.

Let's break this down with definitions of injective and surjective functions:

1. Injective (One-to-One): A function $f: A \rightarrow B$ is injective if different elements in $A$ map to different elements in $B$. Formally, $f(a_1) = f(a_2) \Rightarrow a_1 = a_2$. In other words, no two elements in the domain $A$ have the same image in $B$.

2. Surjective (Onto): A function $f: A \rightarrow B$ is surjective if every element in $B$ has a pre-image in $A$. This means that for every $b \in B$, there exists at least one $a \in A$ such that $f(a) = b$. In simple terms, the function "covers" the entire target set $B$.

3. Bijective (One-to-One Correspondence): A function is bijective if it is both injective and surjective. This implies a perfect "one-to-one" match between elements in $A$ and elements in $B$. Each element in $A$ corresponds to a unique element in $B$, and every element in $B$ has exactly one corresponding element in $A$.

Examples of Bijective Functions

• Identity function: For any set $A$, the function $f: A \rightarrow A$ defined by $f(x) = x$ for all $x \in A$ is bijective, as it pairs each element in $A$ with itself.
• Linear functions with non-zero slope: For real numbers, functions like $f(x) = 2x + 3$ are bijective because they are both injective and surjective on the set of all real numbers.

Properties of Bijective Functions

• Invertibility: A bijective function always has an inverse function, $f^{-1}$, which "undoes" the mapping of $f$. The inverse $f^{-1}: B \rightarrow A$ maps each element of $B$ back to its unique pre-image in $A$.
• Cardinality: If a bijection exists between two sets $A$ and $B$, then $A$ and $B$ have the same number of elements (or the same cardinality, in the case of infinite sets). This is a fundamental concept in set theory, used to compare the sizes of sets.

Would you like more details or have any questions?

Here are some questions to expand your understanding:

1. What are examples of injective functions that are not surjective?
2. What are examples of surjective functions that are not injective?
3. How do we prove that a function is bijective?
4. What role do bijections play in comparing the cardinalities of infinite sets?
5. How is the concept of bijections applied in real-life problem-solving or computer science?

Tip: To verify if a function is bijective, check if it has an inverse—this is only possible if the function is both injective and surjective.