Let $A = \{1, 2, 3, 4\}$, and let $F$ be the set of all functions from $A$ to $A$. Let $S$ be the relation on $F$ defined by:
$\forall f, g \in F$, $fSg \iff f(i) = g(i)$ for some $i \in A$
Let $h: A \to A$ be the function defined by $h(x) = 1$ for all $x \in A$ How many functions $g \in F$ are there so that $gSh$?
So for whatever $x \in A$, $h(x) = 1$. So I want to find how many functions $g \in F$ exist so that $gSh$.
So I made a sort of "recipe" for counting how many functions $g$ exist. Since in order for $gSh$, $g(i) = 1$ for any $i \in A$, since $g(i) = h(i)$ if $gSh$.
Step 1: Map $1 \to 1$
Step 2: Map $2 \to 1$
Step 3: Map $3 \to 1$
Step 4: Map $4 \to 1$
So this is just equal to $1 * 1 * 1 * 1$. So my assumption is that there exists only $1$ function $g$ such that $gSh$. Is this correct? I think I might be misunderstanding the wording, and just want to make sure. Thanks in advance.