In mathematics, an injective function (also known as injection, or one-to-one function) is a function f(x1) ≠ f(x2) (equivalently by contraposition, 1</sub>) {{=}} ''f''(''x''<sub>2</sub>)"}},"i":0}}]}'/>f(x1) = f(x2) implies 1</sub> = ''x''<sub>2</sub>"}},"i":0}}]}'/>x1 = x2). In other words, every element of the function's codomain is the image of 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 f that is not injective is sometimes called many-to-one.
a and a=b; that is, a=b. Equivalently, if f(a)≠f(b) in the contrapositive statement.Symbolically,f:A↣B or X and any subset S→X (which sends any element f:ℝ→ℝ defined by g(x)=x2 is g is redefined so that its domain is the non-negative real numbers exp:ℝ→ℝ defined by x↦lnx is injective.
A proof that a function f(x)=f(y)Proving that functions are injective
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Proof: Let x=y. Therefore, it follows from the definition that f is a linear transformation it is sufficient to show that the kernel of f of a real variable f is injective or one-to-one.
Gallery
edit- X1 and Y2 are subsets of X to map to the same
See also
edit- Bijection, injection and surjection – Properties of mathematical functions
- Injective metric space – Type of metric space
- Monotonic function – Order-preserving mathematical function
- Univalent function – Mathematical concept
Notes
edit- ↑ Sometimes one-one function in Indian mathematical education. "Chapter 1: Relations and functions" (PDF). Archived (PDF) from the original on December 26, 2023 – via NCERT.
- 1 2 3 Farlow, S. J. "Section 4.2 Injections, Surjections, and Bijections" (PDF). Mathematics & Statistics - University of Maine. Archived from the original (PDF) on Dec 7, 2019. Retrieved 2019-12-06.
- ↑ {0,1}→ℝ of the two-element set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction of the real line to the set {0,1}.
- ↑ Halmos, Paul R. (1974), Naive Set Theory, New York: Springer, ISBN 978-0-387-90092-6, p. 38 ff.
