Can someone explain injective, surjective and bijective functions with examples and in his own unique way?
because the concept is too tough.
level is class 12th maths.
If there is any jee aspirant please reply if you have clarity in this concept..
no spam..
points waste naa krna otherwise I will report you.
Answers
Answer:
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
A function maps elements from its domain to elements in its codomain. Given a function {\displaystyle f\colon X\to Y}f \colon X \to Y:
The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection.[1][2] Notationally:
{\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',}{\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',}
or, equivalently (using logical transposition),
{\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').}{\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').}[3][4][5]
The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal. A surjective function is a surjection.[1][2] Notationally:
{\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).}{\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).}[3][4][5]
The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective. A bijective function is also called a bijection.[1][2][3][4][5] That is, combining the definitions of injective and surjective,
{\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),}{\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),}
where {\displaystyle \exists !x}{\displaystyle \exists !x} means "there exists exactly one x".
In any case (for any function), the following holds:
{\displaystyle \forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).}{\displaystyle \forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).}
An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.
A General Function points from each member of "A" to a member of "B".
It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f(x) = 7 or 9" is not allowed)
But more than one "A" can point to the same "B" (many-to-one is OK)
Injective means we won't have two or more "A"s pointing to the same "B".
So many-to-one is NOT OK (which is OK for a general function).
As it is also a function one-to-many is not OK
But we can have a "B" without a matching "A"
Injective is also called "One-to-One"
Surjective means that every "B" has at least one matching "A" (maybe more than one).
There won't be a "B" left out.
Bijective means both Injective and Surjective together.
Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out.
So there is a perfect "one-to-one correspondence" between the members of the sets.
(But don't get that confused with the term "One-to-One" used to mean injective).
Bijective functions have an inverse!
If every "A" goes to a unique "B", and every "B" has a matching "A" then we can go back and forwards without being led astray.
Read Inverse Functions