|Bijection. There is exactly one arrow to every element in the codomain B (from an element of the domain A).|
In mathematics, a bijective function or bijection is a function f : A → B that is both an injection and a surjection. This means: 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.
The term bijection and the related terms surjection and injection were introduced by Nicholas Bourbaki. In the 1930's, he and a group of other mathematicians published a series of books on modern advanced mathematics.
|Not a bijection. (It is not a surjection. It is not an injection.)|
Basic properties[change | change source]
- is a bijective function if there is a unique such that
The element is called the image of the element .
- The formal definition means: Every element of the codomain B is the image of exactly one element in the domain A.
The element is called a pre-image of the element .
- The formal definition means: Every element of the codomain B has exactly one pre-image in the domain A.
Note: Surjection means minimum one pre-image. Injection means maximum one pre-image. So bijection means exactly one pre-image.
Cardinality[change | change source]
- 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[change | change source]
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.)
- The inverse function of the inverse function is the original function.
- A function has an inverse function if and only if it is a bijection.
Note: The notation for the inverse function of f is confusing. Namely,
Examples[change | change source]
Elementary functions[change | change source]
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.)
- Graphic meaning: The function f is a bijection if every horizontal line intersects the graph of f in exactly one point.
- Algebraic meaning: The function f is a bijection if for every real number yo we can find at least one real number xo such that yo=f(xo) and if f(xo)=f(x1) means xo=x1 .
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.)
- Discussion: Every horizontal line intersects a slanted line in exactly one point (see surjection and injection for proofs). Image 1.
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:
- the domain
- the function machine
- the codomain
Example: Suppose our function machine is f(x)=x².
- This machine and domain=ℝ and codomain=ℝ is not a surjection and not an injection. However,
- this same machine and domain=[0,+∞) and codomain=[0,+∞) is both a surjection and an injection and thus a bijection.
Bijections and their inverses[change | change source]
Let f(x):A→B where A and B are subsets of ℝ.
- Suppose f is not a bijection. For any x where the derivative of f exists and is not zero, there is a neighborhood of x where we can restrict the domain and codomain of f to be a bisection.
- The graphs of inverse functions are symmetric with respect to the line y=x. (See also Inverse function.)
Example: The quadratic function defined on the restricted domain and codomain [0,+∞)
- defined by
is a bijection. Image 6: thin yellow curve.
Example: The square root function defined on the restricted domain and codomain [0,+∞)
- defined by
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,+∞)
- defined by
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
- defined by
is the bijection defined as the inverse function of the exponential function: ax. Image 4: thick green curve (a=10).
See also[change | change source]
References[change | change source]
- Weisstein, Eric. "Bijective function" (in English). From MathWorld--a Wolfram Web Resource. http://mathworld.wolfram.com/Bijection.html. Retrieved February 2014.
- C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Bijection" (in English). addison-Wesley. p. 88. http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf. Retrieved February 2014.
- Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics" (in English). Tripod. http://jeff560.tripod.com/i.html. Retrieved February 2014.
- Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. p. 60. ISBN 0-8160-5124-0. (English)
- "Inverse of Bijection is Bijection" (in English). http://www.proofwiki.org/wiki/Inverse_of_Bijection_is_Bijection. Retrieved February 2014.
- "Injection iff Left Inverse" (in English). http://www.proofwiki.org/wiki/Injection_iff_Left_Inverse. Retrieved February 2014.
- "Surjection iff Right Inverse" (in English). http://www.proofwiki.org/wiki/Surjection_iff_Right_Inverse. Retrieved February 2014.
- iff_Left_and_Right_Inverse "Bijection_iff Left and Right Inverse" (in English). http://www.proofwiki.org/wiki/Bijection iff_Left_and_Right_Inverse. Retrieved February 2014.
Other websites[change | change source]
- "Injective, Surjective, Bijective" (in English). 2013. http://www.mathsisfun.com/sets/injective-surjective-bijective.html. Retrieved December 2013. interactive quiz
- "Injectivity, Surjectivity" (in English). Wolfram alpha. https://www.wolframalpha.com/examples/InjectivitySurjectivity.html. Retrieved December 2013. interactive