Injective function

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.
The term "injection" and the related terms "surjection" and "bijection" were introduced by Nicholas Bourbaki. 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 1-1 (read "one-to-one") function. However, this is to be distinguish from a 1-1 correspondence, which is a bijective function (both injective and surjective).
Basic properties.
Formally:
The element formula_5 is called a pre-image of the element formula_6 if formula_7 . Injections have one or none pre-images 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.
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 real-valued function "y"="f"("x") of a real-valued argument "x". (This means both the input and output are real numbers.)
Example: The linear function of a slanted line is 1-1. That is, "y"="ax"+"b" where "a"≠0 is an injection. (It is also a surjection and thus a bijection.)
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.
Example: The quadratic function "f"("x") = "x"2 is not an injection. 
Note: One can make a non-injective 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")="x"² to non-negative numbers (positive numbers and zero). Define 
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.)
Note: The fact that an exponential function is injective can be used in calculations. 
Other examples.
Example: The logarithmic function base 10 "f"("x"):(0,+∞)→ℝ defined by "f"("x")=log("x") or "y"=log10("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 pre-image. Every odd number has no pre-image.