# Mathematics

The Cali Garmo does Math

### Archive

Archive for February, 2014

## Introduction to Group Theory by Ledermann and Weir

Feb 17

Introduction to Group Theory by Ledermann and Weir

Ledermann and Weir take a slightly unique approach in the theory of groups. Their text is slightly difficult to follow in a lot of places as they tend to group things together in non-standard ways. The biggest difference I found with his use of symbols is when talking about homomorphisms. In particular most texts will look at a homomorphism:  and when going from  to  they apply a function such as ; Ledermann and Weird on the other hand choose to use the notation of . Although this format might be more concise, it can at times be a little confusing as to what each symbol is supposed to stand for. My assumption is that he’s trying to show that a homomorphism is similar to group actions with the action being the map , but for an introductory piece, it can be particularly confusing.

All in all I think this book is a really good book for introductory group theory. If you’re willing to invest a little bit of time understanding the notation, you’ll learn more out of this book than most other books on group theory.

## A Mathematical Introduction to Logic by Herbert B. Enderton

Feb 10

A Mathematical Introduction to Logic

I think Enderton does a really nice job introducing logic to those who have never studied it before. Although not always the easiest to follow Enderton lays out all the necessary topics in a nice organised fashion so it’s fairly simple to follow everything. I found his explanation of the pumping lemma to be lacking which made it difficult to follow, but his proof of the compactness theorem more than makes up for it. He does a good introduction of not only proposition and predicate logic, but also goes into second-order logic as well and tackles it the same way as his previous material so it’s easy to follow it all. Although this wasn’t my favourite logic book, it is a nice book to peruse.

## Introduction to the Theory of Computation by Michael Sipser

Feb 3

Introduction to the Theory of Computation

This book was definitely one of the best books I’ve seen for introducing computation theory. Michael Sipser does an amazing job introducing not only Turing Machines, but also different types of machines such as RAM and Finite Automata. He not only gives a good intuitive description and explanation of different machines, but he also does a really good job explaining different languages. Although some of his proofs can be a little difficult to follow, they are all understandable to anyone who is coming at the topic for the first time. He also does a good job explaining the different issues that are relevant in computation theory. unlike most books that I’ve seen so far, he gives a very detailed proof of the Cook-Levin Theorem (That SAT is a NP-Complete language) and also gives multiple examples of proving NP-Completeness which is nice as that seems to be one of the more confusing parts of computation theory for most students. He also gives answers to many of his examples which makes it so that it’s easy to follow on your own and learn everything without needed secondary help.