So I'm struggling right now to understand the concepts behind equivalence relations, equivalence classes, and partitions.
This is what I understand about all these topics right now:
Equivalence Relations: A relation is said to be equivalent if the relation is:
Symmetric, where xRy implies yRx
Transitive, where xRy ^ yRz implies that xRz.
Reflexive, where if x belongs to the real set of numbers, xRx.
Equivalence Classes If we have a equivalence relation represented by ~ on the Set A, where x ∈ A, then we have:
Ex ={y ∈ A:x∼y}, which is a way to represent an equivalence class of x. (What exactly does this mean? Does this mean that x is an equivalence relation of y, and that y belongs to the subset A? Not understanding this, I also don't know how to determine the equivalence class if I've determined that a certain relation is an equivalence relation.)
Partitions A set is said to be a partition if it satisfies the following:
Say we have a Set A, and it has a either finite or infinite amount of subsets B.
The union of all the subsets B should be the set A.
The subsets of B should be disjoint.
Can someone fill in the holes of my understanding? (Some are in italics, also correct any of my understanding if possible)