Right/Left inverse mapping Can someone clearly explain the difference between right and left inverse mapping, if $f: X \to Y$ and $g: Y \to X$ in set theory?
I've read the definitions, but I still have no awareness of what is it, their difference, and how if differs from just inverse mapping.
 A: If you've read the definitions then you already know how they're different.  I guess what you're really looking for is a demonstration.
Define two functions $f,g$ on $\mathbb N$ by $f(n)=2n$ and $g(n)=\lfloor n/2\rfloor$ (that's the floor function of $n/2$, in case you haven't seen it.)  Clearly $g\circ f(n)=n$ for all $n\in\mathbb N$, and that means $g\circ f$ is the identity function on $\mathbb N$. By definition $g$ is a left inverse of $f$, and $f$ is a right inverse of $g$.
But $f\circ g(3)= 2$, so $f\circ g$ is not the identity.  $f$ isn't a left inverse to $g$, and $g$ isn't a right inverse to $f$.
In general when you have two functions $f,g$ such that $f\circ g$ and $g\circ f$ are the identity functions on their respective domains, that's when you say "$g$ and $f$ are inverses of each other" which means that they are both right and left inverses of each other.  The definition of an inverse function is just that it has a counterpart which is both a left and a right inverse.

Note: I really advise not using the phrasing "what is the difference between $X$ and $Y$" when you already do indeed know the definitions are completely different. One could rightly respond to that phrasing by "why do you think they are the same?!"
In a case like this, just ask for examples demonstrating the difference between left inverses and inverse functions. That makes sense because there is a difference to be made.
A: Let $X$ be the set of all integers and $Y$ the set $\{\text{even},\text{odd}\}$ with just two elements.
Suppose $f: X \to Y$ is defined in the obvious way and let
$g: Y \to X$ be
$$
g(\text{even}) = 0 \text{ and  } g(\text{odd}) = 1.
$$
the the composition $f \circ g$ is the identity on $Y$ so $f$ is a left inverse for $g$ and $g$ is a right inverse for $f$.
A: If we have a function $f: X \to Y$ a function $g: Y \to X$ is called a left inverse of $f$ iff $g \circ f = 1_X$, where $1_A: A \to A$ is the identity map on a set $A$ (so $1_A(a)=a$ for all $a$) and it's called a right inverse of $f$ iff $f \circ g = 1_Y$.
The type (left/right) of inverse can be seen by whether you compose with $f$ on the left or the right in the condition giving the identity map.
The map $f: \Bbb N \to \Bbb N$ defined by $f(n)=n+1$ has a left inverse (what is it?) but not a right inverse.
$g$ is an inverse if both a left and a right inverse.
