Problem
The following is a problem from Jacobson's Basic Algebra I:
Let $C$ be a binary relation on $S$. For $r=1,2,3,\dots$ define $C^r=\{(s,t)|\text{ for some } s_1,\dots, s_{r-1}\in S,\text{ one has } sCs, s_1Cs_2,\dots, s_{r-1}Ct\}$. Let $E=1_{S}\cup(C\cup C^{-1})\cup(C\cup C^{-1})^2\cup(C\cup C^{-1})^3\cup\cdots.$ Show that $E$ is an equivalence relation, and that every equivalence relation on $S$ containing $C$ contains $E$. ($E$ is called the equivalence relation generated by $C$.)
My Question
I am having issues with showing that every equivalence relation on $S$ containing $C$ contains $E$. I am also uncertain about my work on the first part.
Partial Answer
An equivalence relation must be reflexive, symmetric, and transitive.
The first set of elements, $1_{S}$, ensures the property of reflexivity via the definition of $1_{S}$: $1_{S}=\{(s,s): s \in S\wedge sCs\}$. Since $E$ is a union of $1_{S}$ and the other sets, all elements in $E$ must satisfy the property that they are in $S$ and related to themselves via $C$.
The second set of elements, $(C\cup C^{-1})$, ensures the property of symmetry. This is because $ \begin{align} (C\cup C^{-1})&=\{(s,t):s,t\in S\wedge sCt\}\cup\{(t,s):t,s\in S\wedge tCs\}\\ &=\{(s,t):s,t \in S\wedge sCt\wedge tCs\}. \end{align} $ In other words, $E$ contains all $(s,t)$ such that $sCt$ and $tCs$.
The rest of the sets of elements, $\bigcup_{i \ge 2}(C\cup C^{-1})^i$, ensures transitivity. I felt this part was difficult because it relies on realizing that the three-part statement of the law of transitivity is an intentional simplification. That is, the statement $aCb \wedge bCc \Rightarrow aCc$ is equivalent to the more long and indefinite statement $aCb_1\wedge b_1Cb_2\wedge b_3Cb_4\wedge \dots \wedge b_nCc \Rightarrow aCc$.
In our case, we have that $\bigcup_{i \ge 2}(C\cup C^{-1})^i$ is the inclusion of all elements such that $sCs_1\wedge s_1Ct$, $sCs_1\wedge s_1Cs_2\wedge s_2Ct$, . . . and so forth. (With, of course, $s,s_1,\dots$ and $t$ in $S$.) This inclusion allows $E$ to be transitive since any given indefinite statement (such as shown above) is true regardless of length.
What I don't understand is how to show that every equivalence relation on $S$ containing $C$ contains $E$. Is this a trick question in the sense that $E$ is the only such equivalence relation on $S$ containing $C$? My understanding of equivalence relations in general is not strong enough to fully envision this question's vast generality.
Thank you for your time.