As pointed out by M. Winter, the converse is not true. My examples have just a few values, but functions usually work on sets with infinitely many elements. In essence, injective means that unequal elements in A always get sent to unequal elements in B. Surjective means that every element of B has an arrow pointing to it, that is, it equals f(a) for some a in the domain of f. A function f : A -> B is said to be onto function if the range of f is equal to the co-domain of f. How to Prove a Function is Bijective without Using Arrow Diagram ? A function that is both One to One and Onto is called Bijective function. The function f is called as one to one and onto or a bijective function, if f is both a one to one and an onto function. And I can write such that, like that. Mathematical Functions in Python - Special Functions and Constants; Difference between regular functions and arrow functions in JavaScript; Python startswith() and endswidth() functions; Hash Functions and Hash Tables; Python maketrans() and translate() functions; Date and Time Functions in DBMS; Ceil and floor functions in C++ $$ Now this function is bijective and can be inverted. If it crosses more than once it is still a valid curve, but is not a function. A function is invertible if and only if it is a bijection. Thus, if you tell me that a function is bijective, I know that every element in B is “hit” by some element in A (due to surjectivity), and that it is “hit” by only one element in A (due to injectivity). Question 1 : The figure shown below represents a one to one and onto or bijective function. Definition: A function is bijective if it is both injective and surjective. Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition for every y in Y there is a unique x in X with y = f(x). Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective. So we can calculate the range of the sine function, namely the interval $[-1, 1]$, and then define a third function: $$ \sin^*: \big[-\frac{\pi}{2}, \frac{\pi}{2}\big] \to [-1, 1]. Each value of the output set is connected to the input set, and each output value is connected to only one input value. Ah!...The beautiful invertable functions... Today we present... ta ta ta taaaann....the bijective functions! The inverse is conventionally called $\arcsin$. More clearly, f maps distinct elements of A into distinct images in B and every element in B is an image of some element in A. Hence every bijection is invertible. Infinitely Many. 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