Injective function

In mathematics, an injective function is a special kind of rule that connects numbers or objects in a very fair way. This rule, also called an injection or a one-to-one function, makes sure that if two things are different, the results after applying the rule will also be different. This means no two different inputs give the same output.

For example, if we have two different numbers, say 3 and 5, and we use an injective function on them, the results will also be different. This is important because it helps us understand how things are linked without any overlaps.

Injective functions are often used when studying algebraic structures like vector spaces. In these areas, an injective function that keeps the structure's rules is called a monomorphism. These ideas help mathematicians explore how different mathematical systems relate to each other.

A function that is not injective is sometimes called many-to-one, meaning more than one input can give the same output. Understanding injective functions helps us see clear and direct relationships in math.

Definition

An injective function, also called a one-to-one function, is a special type of function in mathematics. It means that if you pick two different numbers from the starting group, the function will give you two different results. In other words, no two different inputs will give the same output.

This idea is important because it helps us understand how functions can map one set of numbers to another without mixing things up. Some special symbols are used to show injective functions, like arrows that point in one direction only.

Examples

For visual examples, readers are directed to the gallery section.

• For any set and any smaller group inside it, the way we match each item in the smaller group to itself in the larger group is always injective.
• If a function starts with no items, it is injective.
• If a function starts with just one item, it is always injective.
• The function where we take any number and multiply it by 2 and then add 1 is injective.
• The function where we square a number is not injective because both 1 and -1 give the same result. But if we only use numbers that are zero or more, then it is injective.
• The function where we raise the number e to any power is injective, though it will never give a negative result.
• The function where we take the logarithm of a positive number is injective.
• The function where we take a number to the power n and then subtract the number itself is not injective because both 0 and 1 give the same result.

When talking about functions with all real numbers, an injective function is one where if you draw a horizontal line, it will touch the graph only once. This is called the horizontal line test.

Injections can be undone

Functions with left inverses are always injections. This means if you have a function that maps different inputs to different outputs, you can find another function that "undoes" the first one.

For example, if you have a function that takes a number and changes it in a special way, you can create another function that takes the changed number and brings it back to the original. This shows that some functions can be reversed, even if they are not perfectly two-way matches.

Injections may be made invertible

To change an injective function into one that can be reversed, we can adjust its range to match exactly what the function produces. This makes the function bijective, meaning it can be reversed.

Injective partial functions are known as partial bijections.

inclusion function partial functions partial bijections

Other properties

See also: List of set identities and relations § Functions and sets

• If two functions, f and g, are both injective, then combining them (f after g) is also injective.
• If combining two functions, g after f, is injective, then the first function, f, must be injective. The second function, g, might not be.
• A function f is injective if, whenever two different functions are combined with f and give the same result, those two functions must actually be the same.
• If a function f is injective and A is a smaller group within a larger group X, then you can find A again by applying the opposite of f to the result of f on A.
• If a function f is injective and A and B are both smaller groups within a larger group X, then the combined group of f on A and B is the same as combining the groups A and B first and then applying f.
• Any function can be split into a part that is injective and a part that is "onto" (covers everything). This split is unique in a certain way, and the injective part can be thought of as including the results of the original function as a smaller group within the larger group.
• If a function f is injective, then the group Y (where the function points to) has at least as many members as the group X (where the function starts). If there is also a way to go from Y back to X, then X and Y have exactly the same number of members. (This is known as the Cantor–Bernstein–Schroeder theorem.)
• If both groups X and Y have the same number of members and are small (finite), then a function from X to Y is injective if and only if it also covers every member of Y (in which case it is bijective).
• An injective function that keeps the rules of two math structures the same is called an embedding.
• Unlike covering every member of Y (surjectivity), being injective depends only on how the function behaves, not on Y itself. You can tell if a function is injective by just looking at the pairs of members it connects.

Proving that functions are injective

To show that a function is injective, we use a simple idea: if the function gives the same result for two different inputs, then those inputs must actually be the same. For example, consider the function f(x) = 2x + 3. If f(x) equals f(y), then 2x + 3 equals 2y + 3. This means 2x equals 2y, and so x must equal y. This proves that the function is injective.

There are other ways to prove a function is injective, depending on the type of function. For some functions, showing that a certain value never changes sign can help. In other cases, checking a list of results can show that each input gives a unique output. A picture can also help: if a horizontal line touches the graph of the function at most once, the function is injective.