Bijective function connects elements of two sets such that, it is both one-one and onto function. The elements of the two sets are mapped in such a manner that every element of the range is in co-domain, and is related to a distinct domain element. In simple words, we can say that a function f: A→B is said to be a bijective function or bijection if f is both one-one (injective) and onto (surjective). Show
In this article, we will explore the concept of the bijective function, and define the concept, its conditions, its properties, and applications with the help of a diagram. We will go through various examples based on bijection to better understand the concept. What Is a Bijective Function?A bijective function is a combination of an injective function and a surjective function. Bijective function relates elements of two sets A and B with the domain in set A and the co-domain in set B, such that every element in A is related to a distinct element in B, and every element of set B is the image of some element of set A. The bijective function is both a one-one function and onto function. A bijective function from set A to set B has an inverse function from set B to set A. A bijective function of a set of elements defined to itself is called a permutation. Here every element of the set is related to itself. From the above examples of bijective function, we can observe that every element of set B has been related to a distinct element of set A. The non-bijective functions have some element in set B which do not have a pre-image in set A, or some of the elements in set B is the image for more than one element in set A. Bijective Function DefinitionA function f: A→B is said to be a bijective function if f is both one-one and onto, that is, every element in A has a unique image in B and every element of B has a pre-image in set A. In simple words, we can say that a function f is a bijection if it is both injection and surjection. Bijective Function ConditionsThere are some conditions that need to be satisfied for a function to be a bijection. The bijective functions need to satisfy the following four conditions.
Properties of BijectionNow that we have understood the meaning of bijection, given below are a few important properties of bijective functions which are useful in understanding the concept better:
Injective Surjective BijectiveIn this section, we will discuss the meaning and differences between injective, surjective, and bijective functions. The injective function is also known as the one-one function, and the surjective function is also called the onto function.
One-to-one CorrespondenceOne-to-One functions define that each element of one set called Set (A) is mapped with a unique element of another set called Set (B). A function f : X → Y is said to be one to one correspondence, if the images of unique elements of X under f are unique, i.e., for every x1 , x2 ∈ X, f(x1) = f(x2) implies x1 = x2 and also range = codomain. Otherwise, we call it a non-invertible function or not a bijective function. Therefore we can say, every element of the codomain of one-to-one correspondence is the image of only one element of its domain. Important Notes on Bijective Function
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FAQs on Bijective FunctionBijective function relates elements of two sets such that every element of the domain set is related to a distinct element of the codomain set, and every element of the codomain set has been utilized. How Do We Know If a Function Is a Bijective Function?A function can be easily identified as a bijective function if it is a one-one function, and every element of the codomain set has a preimage in the domain set. Are all Functions Bijections?All the functions are not bijective functions. Some functions can only be injective, or only surjective functions. Some elements of the codomain set may not be utilized or the elements of the codomain set may be related to more than one element of the domain set. How Many Types of Bijective Functions Are There?There is only one bijective function, and it does not have any more classifications. A bijective function is a combination of an injective function and a surjective function. Is x2 a Bijective Function?A square function f(x) = x2 is not a bijective function if the codomain of the function is all real numbers. Is the Inverse of a Bijective Function Bijective?Yes, the inverse of a bijective function is also a bijective function. If a function f: A → B is defined as f(a) = b is bijective, then its inverse f-1(y) = x is also a bijection. How to Prove Bijective Function?To prove that a function is a bijective function, we need to show that every element of the domain has a unique image in the codomain set and each codomain element has a pre-image in the domain set. Why do only Bijective Functions Have Inverses?A function f: A→B has an inverse if and only if it is bijective so that every element of the codomain can be mapped back to an element of the domain that becomes the codomain of the inverse function. What is One-to-one Correspondence?A function f : X → Y is said to be one to one correspondence, if the images of unique elements of X under f are unique, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1 = x2 and also range = codomain. |