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In mathematics, an injective function (also known as injection, or one-to-one function) is a function $f$ that maps distinct elements to distinct elements; that is, $f (x 1) = f (x 2)$ implies $x 1 = x 2$ . (Equivalently, $x 1 \neq x 2$ implies $f (x 1) Templat:\neq f (x 2)$ in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain.^[1] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.^[2] This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function $f$ that is not injective is sometimes called many-to-one.^[1]

Definition

Let $f$ be a function whose domain is a set $X.$ The function $f$ is said to be injective provided that for all $a$ and $b$ in $X,$ if $f(a)=f(b),$ then $a=b$ ; that is, $f(a)=f(b)$ implies $a=b.$ Equivalently, if $a\neq b,$ then $f(a)\neq f(b)$ in the contrapositive statement.

Symbolically, $\forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,$ which is logically equivalent to the contrapositive,^[3] $\forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).$

^ ^a ^b "Injective, Surjective and Bijective". www.mathsisfun.com. Diakses tanggal 2019-12-07.
^ "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Diakses tanggal 2019-12-07.
^ Farlow, S. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu. Diakses tanggal 2019-12-06.

[:0-1] "Injective, Surjective and Bijective". www.mathsisfun.com. Diakses tanggal 2019-12-07.

[2] "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Diakses tanggal 2019-12-07.

[3] Farlow, S. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu. Diakses tanggal 2019-12-06.

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Fungsi
$x \mapsto f (x)$
Contoh domain dan kodomain fungsi
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Kelas/sifat
Konstan Identitas Linear Polinomial Rasional Aljabar Analitik Mulus Kontinu Terukurkan Injektif Surjektif Bijektif
Konstruksi
Pembatasan Komposisi λ Invers
Perumuman
Parsial Bernilai banyak Implisit
l b s