I am confused in the difference between one to one function and one to one correspondence. Please help me out to distinguish between the two. 
Thanks
['https://math.stackexchange.com/questions/63563', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/-1/']
Hope this helps you understand things a bit better. I too had the same question a while ago...  

**Types of Mappings/Functions**  

 a. **Injective mapping (injection)**: **one-to-one mapping** = is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.  

 b. **Surjection**: onto mapping = a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.   

 c. **Bijective mapping (bijection)**: one-to-one and onto mapping = **one-to-one correspondence**   

[NOTE: bijectivity (one-to-one **correspondence**) is a necessary condition for functions to have inverses, whereas injectivity (one-to-one **mapping**) solely will not help in guaranteeing inverses].  

[![Pictorial representations of these mappings](https://i.stack.imgur.com/QdE2r.png)](https://i.stack.imgur.com/QdE2r.png)
1. one to one means injective (a mapping $f$ which maps distinct
elements of its domain to distinct elements of its codomain, i.e. $f(x) \neq f(y)$
whenever $x \neq y$ )
2. one to one correspondence means bijective (a mapping $f$ which is injective AND for every element $y$ in the codomain of $f$, there exists an element of $x$ in the domain of $f$ such that $f(x) = y$ )