mathematical function which is a one-to-one mapping of sets From Wikipedia, the free encyclopedia
In mathematics, a bijective function or bijection is a function f : A → B that is both an injection and a surjection.[1] This is equivalent to the following statement: for every element b in the codomain B, there is exactly one element a in the domain A such that f(a)=b. Another name for bijection is 1-1 correspondence (read "one-to-one correspondence").[2][3]
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Bijection. There is exactly one arrow to every element in the codomain B (from an element of the domain A). |
The term bijection and the related terms surjection and injection 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.
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Not a bijection. (It is not a surjection. It is not an injection.) |
Formally:
where the element is called the image of the element , and the element the pre-image of the element .
The formal definition can also be interpreted in two ways:
Note: Surjection means minimum one pre-image. Injection means maximum one pre-image. So bijection means exactly one pre-image.
Cardinality is the number of elements in a set. The cardinality of A={X,Y,Z,W} is 4. This can be written as #A=4.[5]: 60
By definition, two sets A and B have the same cardinality if there is a bijection between the sets. So #A=#B means there is a bijection from A to B.
Bijections and inverse functions are related to each other, in that a bijection is invertible, can be turned into its inverse function by reversing the arrows.
Formally: Let f : A → B be a bijection. The inverse function g : B → A is defined by if f(a)=b, then g(b)=a. (See also Inverse function.)
Note: The notation for the inverse function of f is confusing. Namely,
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 numbers.)
Proving that a function is a bijection means proving that it is both a surjection and an injection. So formal proofs are rarely easy. Below we discuss and do not prove. (See surjection and injection.)
Example: The linear function of a slanted line is a bijection. That is, y=ax+b where a≠0 is a bijection.
Example: The polynomial function of third degree: f(x)=x3 is a bijection. Image 2 and image 5 thin yellow curve. Its inverse is the cube root function f(x)= ∛x and it is also a bijection f(x):ℝ→ℝ. Image 5: thick green curve.
Example: The quadratic function f(x) = x2 is not a bijection (from ℝ→ℝ). Image 3. It is not a surjection. It is not an injection. However, we can restrict both its domain and codomain to the set of non-negative numbers (0,+∞) to get an (invertible) bijection (see examples below).
Note: This last example shows this. To determine whether a function is a bijection we need to know three things:
Example: Suppose our function machine is f(x)=x².
Let f(x):A→B where A and B are subsets of ℝ.
Example: The quadratic function defined on the restricted domain and codomain [0,+∞)
is a bijection. Image 6: thin yellow curve.
Example: The square root function defined on the restricted domain and codomain [0,+∞)
is the bijection defined as the inverse function of the quadratic function: x2. Image 6: thick green curve.
Example: The exponential function defined on the domain ℝ and the restricted codomain (0,+∞)
is a bijection. Image 4: thin yellow curve (a=10).
Example: The logarithmic function base a defined on the restricted domain (0,+∞) and the codomain ℝ
is the bijection defined as the inverse function of the exponential function: ax. Image 4: thick green curve (a=10).
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