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Injective function
Injection. Maximum one arrow to each element in the codomain B (from an element in domain A). 
In mathematics, a injective function is a function f : A → B with the following property. For every element b in the codomain B, there is at most one element a in the domain A such that f(a)=b, or equivalently, distinct elements in the domain map to distinct elements in the codomain.^{[1]}^{[2]}^{[3]}
The term injection and the related terms surjection and bijection were introduced by Nicholas Bourbaki.^{[4]} In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.
An injective function is often called a 11 (read "onetoone") function. However, this is to be distinguish from a 11 correspondence, which is a bijective function (both injective and surjective).^{[5]}
Not an injection. Two elements {X} and {Y} in the domain A are mapped to the same element {1} in the codomain B. 
Contents
Basic properties
Formally:
 [math]f:A \rightarrow B[/math] is an injective function if [math]\forall a_1, \,a_2 , \in A, \,\,\,\, a_1 \ne a_2 \,\, \Rightarrow \,\, f(a_1) \ne f(a_2)[/math] or equivalently
 [math]f:A \rightarrow B[/math] is an injective function if [math]\forall a_1, \,a_2 , \in A, \,\,\,\,f(a_1)=f(a_2) \,\, \Rightarrow \,\, a_1=a_2[/math]^{[2]}
The element [math]a[/math] is called a preimage of the element [math]b[/math] if [math]f(a)=b[/math] . Injections have one or none preimages for every element b in B.
Cardinality
Cardinality is the number of elements in a set. The cardinality of A={X,Y,Z,W} is 4. This is written as #A=4.^{[6]}
If the cardinality of the codomain is less than the cardinality of the domain, then the function cannot be an injection. For example, there is no injection from 6 elements to 5 elements, since it is impossible to map 6 elements to 5 elements without a duplicate.
Examples
Elementary functions
Let f(x):ℝ→ℝ be a realvalued function y=f(x) of a realvalued argument x. (This means both the input and output are real numbers.)
 Graphic meaning: The function f is an injection if every horizontal line intersects the graph of f in at most one point.
 Algebraic meaning: The function f is an injection if f(x_{o})=f(x_{1}) means x_{o}=x_{1}.
Example: The linear function of a slanted line is 11. That is, y=ax+b where a≠0 is an injection. (It is also a surjection and thus a bijection.)
 Proof: Let x_{o} and x_{1} be real numbers. Suppose the line maps these two xvalues to the same yvalue. This means a·x_{o}+b=a·x_{1}+b. Subtract b from both sides. We get a·x_{o}=a·x_{1}. Now divide both sides by a (remember a≠0). We get x_{o}=x_{1}. So we have proved the formal definition and the function y=ax+b where a≠0 is an injection.
Example: The polynomial function of third degree: f(x)=x^{3} is an injection. However, the polynomial function of third degree: f(x)=x^{3} –3x is not an injection.
 Discussion 1: Any horizontal line intersects the graph of f(x)=x^{3} exactly once. (Also, it is a surjection.)
 Discussion 2. Any horizontal line between y=2 and y=2 intersects the graph in three points so this function is not an injection. (However, it is a surjection.)
Example: The quadratic function f(x) = x^{2} is not an injection.
 Discussion: Any horizontal line y=c where c>0 intersects the graph in two points. So this function is not an injection. (Also, it is not a surjection.)
Note: One can make a noninjective function into an injective function by eliminating part of the domain. We call this restricting the domain. For example, restrict the domain of f(x)=x² to nonnegative numbers (positive numbers and zero). Define
 [math]f_{/[0,+\infty)}(x):[0,+\infty) \rightarrow \mathbf{R}[/math] where [math]f_{/[0,+\infty)}(x) = x^2[/math]
This function is now an injection. (See also restriction of a function.)
Example: The exponential function f(x) = 10^{x} is an injection. (However, it is not a surjection.)
 Discussion: Any horizontal line intersects the graph in at most one point. The horizontal lines y=c where c>0 cut it in exactly one point. The horizontal lines y=c where c≤0 do not cut the graph at any point.
Note: The fact that an exponential function is injective can be used in calculations.
 [math]a^{x_0}=a^{x_1} \,\, \Rightarrow \,\, x_0=x_1, \, a\gt 0 [/math]
 Example: [math]100=10^{x3} \,\, \Rightarrow \,\, 2=x3 \,\, \Rightarrow \,\, x=5[/math]
Other examples
Example: The logarithmic function base 10 f(x):(0,+∞)→ℝ defined by f(x)=log(x) or y=log_{10}(x) is an injection (and a surjection). (This is the inverse function of 10^{x}.)
Example: The function f:ℕ→ℕ that maps every natural number n to 2n is an injection. Every even number has exactly one preimage. Every odd number has no preimage.
Related pages
References
 ↑ "The Definitive Glossary of Higher Mathematical Jargon" (in enUS). 20190801. https://mathvault.ca/mathglossary/.
 ↑ ^{2.0} ^{2.1} Weisstein, Eric W.. "Injection" (in en). https://mathworld.wolfram.com/Injection.html.
 ↑ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Onto Mapping" (in English). AddisonWesley. p. 568. http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf. Retrieved 20140101.
 ↑ Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics" (in English). Tripod. http://jeff560.tripod.com/i.html. Retrieved 20140201.
 ↑ Weisstein, Eric W.. "OnetoOne" (in en). https://mathworld.wolfram.com/OnetoOne.html.
 ↑ Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. p. 60. . (in English)
Other websites
 "Injective, Surjective, Bijective" (in English). 2013. http://www.mathsisfun.com/sets/injectivesurjectivebijective.html. Retrieved 20131201. interactive quiz
 "Injectivity, Surjectivity" (in English). https://www.wolframalpha.com/examples/InjectivitySurjectivity.html. Retrieved 20131201. interactive
