Home > Injective function

More formally, a function f: X → Y is injective 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, f is injective if, for every x and x' in X, whenever f(x) = f(x'), we must have x = x'.

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.)

1 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: [0, ∞) → 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'.

2 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.
• By definition, 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 categoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan SetCategory theory In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set . of sets.
• If f: X → Y is injective and A is a subsetIf X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes X; Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equa 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 numberAlternative meaning: number of pitch classes in a set. In linguistics, cardinal numbers is the name given to number words that are used for quantity one two three , as opposed to ordinal numbers, words that are used for order first second third . See Hows.
• If both X and Y are finiteIn mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,. n with n isin N . It is a theorem that a set is finite if and only if there exists no bijection between the set and any of its prop with the same number of elements, then f : X → Y is injective if and only if f is surjective.