Abstract Algebra Learning Log - Preliminaries
Recently, I have been working on Abstract Algebra which has derived category theory and exposes its strong relation with computer science.
All these learning logs will be based on one online and free textbook (2019 Version), naming Abstract Algebra which is licensed under the GNU FDL.
This learning log is basically in two parts, one is my own understandings or complementions to some of the parts, another part is my own answers to part of the Exercises section in the end of the chapter. And succeeding learning logs will mainly based on this form too.
First part may contain works in the textbook itself, therefore, this series of posts is licensed under the GNU FDL either.
Understandings and complementions to textbook
Page 6, Proof of De Mogran's Laws
We define two statements, and , representing and respectively. statement . (The relation between and can be deduced by simply enumerating through all the possible values of and , e.g. or for statements). Thus, . Conversely, we can prove . Therefore, . 2. Similarly, we can deduce it by using proposition logic.
Page 12, Proof of implications between partition and relation
Given an equivalence relation on a set , the equivalence classes of form a partition of , then there is an equivalence relation on with equivalence classes . Conversely, if partition of a set , then there is an equivalence relation on with equivalence classes .
By using the reflective property, we have . Thus, by the definition of the equivalence class. Therefore, the .
Now, we should show , either or .
Suppose , Then, .
Conversely, we can also show in the same way. Thus, the and are either disjoint or completely the same.
Now we need to show that given a partion on , we can construct a relation on such that partion is the equivalence classes of .
Suppose we have a relation . We define . So we should show that the relation is equivalent.
Reflectivity since , will be in the same as itself does, so it is trivial to prove.
- Transitivity If , then and are in the same part. If , than and are in the same part. Thus and are in the same part, which is .
Thus relation is equivalence relation. Since no element can be both in two parts, where .
Now we need to prove the set of equivalence classes is the same as partition .
If there is where , then there must be some in other equivalence class or in other part of . Both of which above have contradicted the definition of . And the will be the same as the set of equivalence classes.
My Answers to part of Exercises section .
Disclaimer: this part is my own work to the section, thus no guarantees is provided in terms of correctness. You may use it neither as instruction or correct answer
General solution to problems except and
All the proof-based problems involving the equality of operation of set theory (limited to union and intersect only) can be proved by transforming both sides into statements and enumerating through all the possible values of single statements, which construct the left-hand-side or right-hand-side statement, to show that set on each side is a subset of the other. For example, Proof of De Morgan's laws.
In order to let be a mapping which is well-defined, we need to make sure that where and .
- Only if implies the above implication. Thus it isn't a mapping
- It is a mapping since it fulfills the requirement above.
- Only if or . Not a mapping.
- It is a mapping.
(d) We have to show that in order to prove that is one-to-one. Suppose . Since is onto, there exists such and that and . So can be written as . Given is one-to-one, we know that . Thus , resulting in .
Suppose there is a element such that is not related to anything. Then transitivity and symmetry hold but reflective holds since we need to have .
Other parts I find not worth sharing or taking notes.remain unsolved.
I would prefer a quick and informal one rather than a formal one.
- Abstract Algebra
- Equivalence Class Question - Reddit
My math teacher.