What did I find difficult?
A relation R on a set A is called an equivalence relation if R is reflexive, symmetric, and transitive.
So, if a relation has all three of these properties, then it is an equivalence relation.
An equivalence class is the class of all members of a set that are in a given equivalence relation.
I found equivalence classes to be relatively simple to understand. Reading through the example problems did not give me much trouble. I imagine that as we get more into this, I will come across more difficult applications of equivalence relations.
On the other hand, reading through the properties of equivalence classes was difficult for me. I understand what an equivalence class is (at least I think I do), but in proving different properties of equivalence classes, I'm having trouble.
What did I find interesting?
I find the concept of equivalence classes interesting. I'm still having trouble understanding 8.4, so I'm going to focus on 8.3. It seems to be intuitive that if R is reflexive, symmetric, and transitive then it is an equivalence relation. For the equivalence class, I like this [a] = {x \in A : x R a} My understanding is that this consists of all elements that are related to a.
It made sense when it said, loosely speaking, that [a] consists of the relatives of a. It also makes sense that the equivalence classes form a partition of the set.
I feel that this is slightly different from other chapters, but I like the way of thinking. I can see how this will be helpful later on, maybe in proving parts of a statement in order to prove the whole thing true.
No comments:
Post a Comment