What does it mean to "determine" an equivalence relation? I don't understand the following problem:

What does it mean exactly that a number of pairs can "determine" an equivalence relation? Say if I have the following set: {1, 2, 3}, and a relation R that's true for (a, b) if a=b. Then would the pairs {1, 1}, {2, 2}, and {3, 3} "determine" this relation? How then, does the author arrive at n/2 for this solution?
This problem is from this problem set.
 A: The author seems to describe very carefully what it means for a collection of pairs to "determine" a relation. You start with a relation that you know is an equivalence relation, and then start throwing things out until you cannot throw out anymore.
For instance, if you begin with the relation
$$
R = \{(1,1), (2,2), (3,3), (2,3), (3,2)\}
$$
Then you can throw out $(2,2)$ and $(2,3)$ and only keep $(3,2)$
$$
R'  = \{(1,1), (3,3), (3,2)\}
$$
because once you know $(3,2) \in R$, you know that $(2,3) \in R$ by symmetry and you know $(2,2) \in R$ by transitivity.
The author seems to arrive at $n/2$ as follows : You need each element of $S = \{1,2,\ldots, n\}$ to be present in some pair. We can order the elements so that
$$
R' = \{(1,2), (3,4), \ldots, (n-1,n)\} \text{ (assume $n$ even for now)}
$$
Now, $|R'| = n/2$, and this is the bare minimum. If any element appears in two pairs, then the element that it "kicked out" would have to show up in another pair, thereby increasing the number of pairs.
One has to make this argument rigorous, but that is the gist of it.
A: This is explained, although in a slightly roundabout way, at the beginning of Problem 4 on the problem set.
Rephrasing the definition, hopefully slightly more clearly: if $R$ is any set of pairs, then the equivalence relation it determines is a relation $E_R$ on the set $\bigcup R$ (i.e. the set of all elements that appear in any pair in $R$); $E_R$ is defined as the smallest equivalence relation on $\bigcup R$ containing $R$.
Exercise: for $(x,y) \in \bigcup R$, you have $(x,y) \in E_R$ if and only if there is some sequence $(x_0,x_1,\ldots,x_n)$ such that $x_0 = x$, $x_n = y$, and for each $0 \leq i < n$, either $(x_i,x_{i+1}) \in R$ or $(x_{i+1},x_i) \in R$.
The slightly nonstandard part of the definition is taking the domain of the equivalence relation to be also determined by $R$, rather than as given separately.  A more common version of the definition is: suppose $X$ is a set, and $R \subseteq X \times X$; then the equivalence relation on $X$ generated by $R$ is the smallest equivalence relation on $X$ containing $R$. (And again, one can give an explicit definition of this by sequences of elements that are related in $R$.)
