Question

explain Injection function, Surjection function and Bijection function​

Answer 1

Answer:

ok

Step-by-step explanation:

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain.

Answer 2

Answer:

• 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. 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').}

• 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 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).}

• The function is bijective (one-to-one and onto or one-to-one correspondence) 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. 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.