Injection (mathematics)

A function f: X → Y is called injective or one-to-one or an injection if for every y in the codomain Y there is at most one x in the domain X with f(x) = y. Put another way, given x and x' in X, if f(x) = f(x'), then it follows that x = x'.

Surjective, not injective	Injective, not surjective
Bijective	Not surjective, not injective

When X and Y are both the real line R, then an injective function f: R → R can be visualized as one whose graph is never intersected by any horizontal line more than once. (This is the horizontal line test.)

Examples and counterexamples

Consider the function f: R → R defined by f(x) = 2x + 1. This function is injective, since given arbitrary real numbers x and x', if 2x + 1 = 2x' + 1, then 2x = 2x', so x = x'.

On the other hand, the function g: R → R defined by g(x) = x² is not injective, because (for example) g(1) = 1 = g(−1).

However, if we define the function h: R⁺ → R by the same formula as g, but with the domain restricted to only the nonnegative real numbers, then the function h is injective. This is because, given arbitrary nonnegative real numbers x and x', if x² = x'², then |x| = |x'|, so x = x'.

Properties

• A function f: X → Y is injective if and only if X is the empty set or there exists a function g: Y → X such that g ^o f  equals the identity function on X.
• A function is bijective if and only if it is both injective and surjective.
• If g ^o f is injective, then f is injective.
• If f and g are both injective, then g ^o f is injective.
• 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 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.

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See also: Surjection, Bijection