When do two permutations commute?

Question

Let $\sigma$ $\in$ $S_n(Symmetric\;Group)$ .Let $S_\sigma$ := [i $\in$ {1,2,....,n} | $\sigma(i)=i$] .I have to show that if $\phi$ and $\tau$ are two disjoint cycles, then $\phi^i$ and $\tau^j$ are disjoint . Also show that if $\sigma,\tau\in S_n$ are disjoint then $\sigma\cdot\tau = \tau\cdot\sigma$ , For the second case I can show that if $\sigma,\tau$ are disjoint cycles then commutative property holds. But for general $\sigma,\tau$ how can I show that? Please give solution for the first one.$\\$ EDIT: Two permutaions $\sigma,\tau\in S_n$ are called disjoint if $S_\sigma^c \cap S_\tau^c=\emptyset$.