Injective function Totally Explained

Everything about Injection Mathematics totally explained

In mathematics, an injective function is a function which associates distinct arguments with distinct values.
An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (the latter isn't to be confused with one-to-one correspondence, for example a bijective function).
A function f that isn't injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", for example each argument is mapped to at most one value.)

Definition

Let f be a function whose domain is a set A. It is injective if, for all a and b in A such that f(a)=f(b), we've a = b.

Examples and counter-examples

For any set X, the identity function on X is injective.
The function f : R → R defined by f(x) = 2x + 1 is injective.
The function g : R → R defined by g(x) = x² is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the non-negative real numbers [0,+∞),then g is injective.
The exponential function $exp : mathbb in$ R will be injective but have no left inverse).
Note that g may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity on Y. In other words, a function that can be undone or "reversed", such as f, isn't necessarily invertible (bijective). Injections are "reversible" but not always invertible.
Although it's impossible to reverse a non-injective (and therefore information-losing) function, you can at least obtain a "quasi-inverse" of it, that's a multiple-valued function.
Injections may be made invertible
In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as incl_J,Yog, where incl_J,Yis the inclusion function from J into Y.

Other properties

If f and g are both injective, then f o g is injective.

If g o f is injective, then f is injective (but g need not be).

f : X → Y is injective if and only if, given any functions g, h : W → X, whenever f o g = f o h, then g = h. In other words, injective functions are precisely the monomorphisms in the category Set of sets.

If f : X → Y is injective and A is a subset of X, then f⁻¹(f(A)) = A. Thus, A can be recovered from its image f(A).

If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).

Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h.

If f : X → Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal numbers.

If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective.

Every embedding is injective.Further Information

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Everything about Injection Mathematics totally explained

Definition

Examples and counter-examples

Injections may be made invertible

Other properties