Let $S \subseteq \mathbb{Z}$, and define a relation $R$ on $S \times S$ by
$$(m, n)R(s, t) \quad \text{ if and only if } \quad m + n = s + t$$
Consider the set $S = \{1, 2, 3, 4, 5, 6\}$. How many equivalence
classes does $R$ have? Describe the equivalence classes of $R$
without explicitly listing the partition of $S × S$.
My try:
I have proved that the relation is an equivalence relation, $R$ is reflexive, symmetric, and transitive. For the set $S$ there are $2^6$ subsets.
Any help starting this problem would be appreciated!
Best Answer
Two elements $(a,b)$ and $(c,d)$ in $S \times S$ are related if $a+b=c+d$. For example, if we take the element $(1,5) \in S \times S$, then $(1,5) \sim (1,5)$ because $1+5=1+5=6$.
So for finding the equivalence class of $(1,5)$ we ask ourselves: what are all other elements $(c,d) \in S \times S$ such that $(1,5) \sim (c,d)$?
For that, we want $c+d=6$. So look for all the pairs that satisfy this condition. $$[(1,5)]=\{(1,5), (5,1), (3,3), (2,4), (4,2)\}.$$ Likewise $$[(1,1)]=\{(1,1)\} \qquad \text{and} \qquad [(5,6)]=\{(5,6), (6,5)\}.$$
Hopefully you can proceed from here to get the remaining equivalence classes.
Note: If you just want the number of equivalence classes (without describing them), then note that each equivalence class can be associated with the sum of the pairs in that, e.g. the class $[(1,5)]$ can be associated to the sum $6$ and class $[(1,1)]$ can be associated with the sum $2$ and so on. So the number of distinct classes is the number of distinct sums.