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.)
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) = x^{2} 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 x^{2} = x'^{2}, then |x| = |x'|, so x = x'.
- 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.
See also: Surjection, Bijection
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