Let $A = \{1, 2, 3, ... , n\}$ where $n$ is a positive integer. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by:
for all $g, f \in F, fRg$ if and only if $f(i) \lt= g(i)$ for some $i \in A.$ Let $I$A : $A \to A$ be the identity function on $A$ defined by $I$A$(x)$ = $x$ for all $x \in A$.
How many elements $f \in F$ are there so that $I$A$Rf$? Explain.
My attempt: We know that identity function reverses what the function did. So, since identity function here is defined as $I$A$(x)$ = $x$, then there are n elements?
Hint: $I_A\,R\,f$ means that there is some $i\in A$ with $i = I_A(i) \le f(i)$. So the functions $f$ such that $I_A\,R\,f$ is false are those functions $f\in F$ such that $i>f(i)$ for all $i\in A$. How many such functions are there?