Functions: Injective, Surjective, Bijective

Functions: Injective, Surjective, Bijective

In mathematics, a function is a relationship between two sets where every element of the domain is associated with exactly one element of the codomain. Functions play a crucial role in many branches of mathematics, such as algebra, calculus, and set theory. When discussing functions, it is important to understand the concepts of injective, surjective, and bijective functions. These terms describe different types of mappings between sets based on how elements of the domain are related to elements of the codomain.

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1. Definition of a Function

A function $f$ from a set $A$ (called the domain) to a set $B$ (called the codomain) is a rule that assigns to each element $a \in A$ exactly one element $b \in B$. This is written as:

$$f: A \to B$$

where $f(a) = b$ means that the function $f$ maps the element $a$ to the element $b$.

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2. Injective Functions (One-to-One)

A function $f: A \to B$ is injective (or one-to-one) if different elements in the domain map to different elements in the codomain. In other words, if $f(a_1) = f(a_2)$, then it must follow that $a_1 = a_2$.

Formal Definition:

A function $f: A \to B$ is injective if:

$$\forall a_1, a_2 \in A, \, f(a_1) = f(a_2) \implies a_1 = a_2.$$

Interpretation:

An injective function ensures that no two distinct elements in the domain map to the same element in the codomain. Each element in the codomain can be "hit" at most once by the function.

Example of Injective Function:

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x$, where $\mathbb{R}$ denotes the set of real numbers.

• If $f(x_1) = f(x_2)$, then $2x_1 = 2x_2$, which implies $x_1 = x_2$.
• Thus, $f$ is injective because no two different values of $x$ map to the same output.

Non-Example (Not Injective):

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$.

• For $x_1 = 2$ and $x_2 = -2$, we have $f(2) = f(-2) = 4$, but $2 \neq -2$.
• Therefore, $f(x) = x^2$ is not injective because two different values in the domain map to the same value in the codomain.

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3. Surjective Functions (Onto)

A function $f: A \to B$ is surjective (or onto) if every element in the codomain has at least one element in the domain that maps to it. In other words, the function "covers" the entire codomain.

Formal Definition:

A function $f: A \to B$ is surjective if:

$$\forall b \in B, \, \exists a \in A \text{ such that } f(a) = b.$$

Interpretation:

A surjective function ensures that every element of the codomain is "reached" by the function, meaning that the function maps to the entire codomain. Some elements in the domain might map to the same element in the codomain.

Example of Surjective Function:

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$.

• For every $b \in \mathbb{R}$, we can find an $a \in \mathbb{R}$ such that $f(a) = a^3 = b$. Specifically, $a = \sqrt[3]{b}$.
• Therefore, $f(x) = x^3$ is surjective because every real number is the cube of some real number.

Non-Example (Not Surjective):

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$.

• The function cannot produce negative numbers because $x^2 \geq 0$ for all $x \in \mathbb{R}$. Therefore, there is no $x \in \mathbb{R}$ such that $f(x) = -1$.
• Hence, $f(x) = x^2$ is not surjective when the codomain is $\mathbb{R}$, because it does not "hit" all values in the codomain.

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4. Bijective Functions (One-to-One Correspondence)

A function $f: A \to B$ is bijective if it is both injective and surjective. In other words, a bijection is a function where every element of the domain is mapped to a unique element in the codomain, and every element of the codomain is mapped to some element in the domain.

Formal Definition:

A function $f: A \to B$ is bijective if:

• $f$ is injective: $\forall a_1, a_2 \in A, f(a_1) = f(a_2) \implies a_1 = a_2$,
• $f$ is surjective: $\forall b \in B, \exists a \in A \text{ such that } f(a) = b$.

Interpretation:

A bijective function establishes a one-to-one correspondence between the elements of the domain and the elements of the codomain. Each element of the domain is paired with exactly one element of the codomain, and each element of the codomain is paired with exactly one element of the domain.

Example of Bijective Function:

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x + 3$.

• Injectivity: If $f(x_1) = f(x_2)$, then $2x_1 + 3 = 2x_2 + 3$, which simplifies to $x_1 = x_2$.
• Surjectivity: For any $b \in \mathbb{R}$, there exists an $a = \frac{b - 3}{2} \in \mathbb{R}$ such that $f(a) = b$.
• Therefore, $f(x) = 2x + 3$ is bijective.

Non-Example (Not Bijective):

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$.

• This function is not injective because $f(2) = f(-2) = 4$, but $2 \neq -2$.
• It is also not surjective because it cannot map to negative numbers.
• Thus, $f(x) = x^2$ is not bijective when the domain and codomain are $\mathbb{R}$.

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5. Summary of Function Types

• Injective (One-to-One): A function is injective if no two distinct elements in the domain map to the same element in the codomain. Formally: $f(a_1) = f(a_2) \implies a_1 = a_2$.
• Surjective (Onto): A function is surjective if every element in the codomain is mapped to by at least one element in the domain. Formally: $\forall b \in B, \exists a \in A$ such that $f(a) = b$.
• Bijective (One-to-One Correspondence): A function is bijective if it is both injective and surjective. There is a one-to-one correspondence between the domain and codomain.

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6. Practical Examples

• Injective but not Surjective: The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x$. It’s injective, but not surjective since not all real numbers can be reached (e.g., no $x$ maps to an odd number).
• Surjective but not Injective: The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$. It’s surjective when the codomain is $[0, \infty)$, but not injective because two different values (e.g., 2 and -2) can map to the same result.
• Bijective: The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x + 3$. It is both injective and surjective, hence bijective.

Understanding these properties helps in determining the structure and behavior of functions in various mathematical contexts.