Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is a fixed integer.
Define a relation $R$ on $\mathscr F$ by : for any $f,g \in \mathscr F$, $\;fRg\;$ if and only if $f - g$ is a constant function (i.e. there is a constant $c$ so that $f(x) - g(x) =c$ for all $x\in A$).
Let $f_1\in \mathscr F$ be defined by: $f_1(1)=2,f_1(2)=n,f_1(3)=1$. Suppose that $g \in \mathscr F$ is arbitrary so that $gRf_1$. Prove that $g=f_1$ and thus $[f_1]$ is just $ \{f_1\}$.
My work ( not much):
so $gRf_1$ means $g - f_1 = c$ $\Rightarrow g = f_1+c.$ To prove $g=f_1$ I need to prove $g\subseteq f_1$ and $f_1\subseteq g$. So for $g\subseteq f_1$ , let $x\in g$ and I need to prove $x\in f_1$.
So I am not sure where to go from here.
