For a permutation p: X → X, let $p^{k}$ denote the permutation arising by a k-fold composition of p, i.e. $p^{1} = p \ and \ p^{k} = p◦p^{k−1}$. Define a relation ≈ on the set X as follows: i ≈ j if and only if there exists a k ≥ 1 such that $p^{k}(i) = j$. Prove that ≈ is an equivalence relation on X, and that its classes are the cycles of p.
I am new to this, what does it mean by its classes? does it mean its elements in the set X? I do see the connection between the k-fold composition and the permutation of p. I think I just do not understand the language. Please give me some hints.
An equivalence relation is a relation that is reflexive ($i \approx i \;\forall i$), symmetric $(i \approx j \iff j \approx i)$, and transitive $(i \approx j$ and $j \approx k \implies i \simeq k$). Because of those axioms, an equivalence relation induces a partition of the set $X$ into subsets of elements which are all equivalent to each other. That is, if we look at the sets $Y_i = \{j \in X : j \approx i\}$ for various $i \in X$, then each pair $Y_i$ and $Y_j$ are either the same set or disjoint from each other. The subsets $Y_i$, more commonly denoted $[i]$, are called the "equivalence classes" for the relation $\approx$. $[i]$ is called "the equivalence class of $i$".