Let $f: \{1, 2, 3\} \rightarrow \{a, b, c\}$ be a one-one and onto function given by $f(1) = a$,$f(2) = b$,and $f(3) = c$. Show that there exists a function $g: \{a, b, c\} \rightarrow \{1, 2, 3\}$ such that $g \circ f = I_X$ and $f \circ g = I_Y$,where $X = \{1, 2, 3\}$ and $Y = \{a, b, c\}$.

(A) Define a function $g: \{a, b, c\} \rightarrow \{1, 2, 3\}$ such that $g(a) = 1$,$g(b) = 2$,and $g(c) = 3$.
Now,consider the composite function $g \circ f: X \rightarrow X$:
$(g \circ f)(1) = g(f(1)) = g(a) = 1$
$(g \circ f)(2) = g(f(2)) = g(b) = 2$
$(g \circ f)(3) = g(f(3)) = g(c) = 3$
Since $(g \circ f)(x) = x$ for all $x \in X$,$g \circ f = I_X$.
Next,consider the composite function $f \circ g: Y \rightarrow Y$:
$(f \circ g)(a) = f(g(a)) = f(1) = a$
$(f \circ g)(b) = f(g(b)) = f(2) = b$
$(f \circ g)(c) = f(g(c)) = f(3) = c$
Since $(f \circ g)(y) = y$ for all $y \in Y$,$f \circ g = I_Y$.