Let $A = \{1, 2, 3, 4\}$ and let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by "For all $f, g$ in $F$, $(f, g)$ in $R$ if and only if $f (i) = g (i)$ for some $i$ in $A$."
Let $N, S$ in $F$ be the functions defined by $N(x) = 1$ and $S(x) = 2$ for all $x$ in $A$.
Is $R$ reflexive? Symmetric? Transitive? Please help me through this problem, I don't think it's either because there is no way $N(x) = S(x)$, but it's obviously not right.
Write $f\sim g$ if $(f,g)\in R$.
For reflexivity of $R$, you need to check whether $f\sim f$ for all $f\in F$.
For symmetry of $R$, you need to check whether, for all $f,g\in F$, it holds that $f\sim g$ implies $g\sim f$.
For transitivity of $R$, you need to check whether $f\sim g$ and $g\sim h$ implies $f\sim h$ for arbitrary $f,g,h\in F$.
Since this looks like home work, I will refrain from posting the concrete answer.