Function ( Onto ) (one-to-one)

Surjective (onto) and Injective (one-to-one) functions :



Onto (Subjective):

Definition : 
A function f  from A to B is called Onto , or surjective , if and only if for every element b belong to B there is an element a belong to A with f(a)=b. A function f is called a surjection if it is Onto .

Remark :
A function f  is onto if all y exist x (f(x)=y) , where the domain for x is the domain of the function and the domain for y is the codomain of the function.

Note :
number elements of Rang = number elements of Codomain

One-to-One (Injective):  

Definition : 

A function f is said to be one-to-one , or injective , if and only if f(a)=f(b) implies that a=b for all a and b in the domain of f . A function is said to be an injection if it is one-to-one.
Note that a function f  is one-to-one if and only if f(a)!=f(b) wherever a!=b .This way of expressing that f is one-to-one  is obtained by taking the contrapositive of the implication in the definition .

Remark :
We can express that f is one-to-one using quantifiers as All a All b (f(a)=f(b) --> a=b) or  equivalently All a All b (f(a)!=f(b) --> a!=b ) , where the universe of discourse is the domain of the function .


Note :
number elements of Rang = number elements of Domain