I am currently self-learning Modern Algebra on MIT OCW. I am at the lecture on cosets and am stuck on the theorem connecting equivalence relation and partition. I understand what the theorem implies, but have difficulty making sense of the sentence: "the data of an equivalence relation is the same as the data of a partition".
First of all, what does "data" mean in this context? It cannot possibly refer to the elements can it? I will provide an example.
Let S be the set {$x_1, x_2, x_3, x_4, x_5$}. The equivalence relation R that induces the partition {{$x_1, x_2$}, {$x_3, x_4$}, {$x_5$}} is $R = R_1 ∪ R_2 ∪ R_3$, where R_1 = {$(x_1, x_1), (x_1, x_2), (x_2, x_2), (x_2, x_1)$} R_2 = {$(x_3, x_3), (x_3, x_4), (x_4, x_4), (x_4, x_3)$} R_3 = {$(x_5, x_5)$}. What would the data be for this equivalence relation R and what would the data be for this partition?
Thank you!
The sentence "the data of an equivalence relation is the same as the data of a partition" means that, given an equivalence relation, you can exactly construct the blocks of a partition, and vice versa (Think about it!). That is, the "information" you learn by being given a partition is exactly the same as the "information" you learn by being given an equivalence relation.
Once you get this notion of "data", your example should make sense.