February 2021
I spent the last few weeks learning how to read and write basic math proofs. It was slightly challenging, but at the end it was highly rewarding, definitely one of the highest ROI activity I've done in my life. The book that I used was Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al.
The book is organized into two parts. The first part discusses basic mathematical concepts such as sets, functions, relations, cardinality, as well as introducing a host of proof techniques. The second half of the book comprises of a collection of proof examples in a variety of math subfields, including number theory, calculus, linear algebra, and more.
I went through the first half of the book in roughly a month's time, averaging about 3 hours daily. I studied all the examples in these first 11 chapters, and also attempted to do all the odd numbered exercises at the end of each chapter. I managed to complete about 90% of these exercises.
The chapter on set cardinality was the most challenging. I skipped through subsection 11.5, which discusses the Schröder–Bernstein theorem. I also did not go through the collection of proof examples in the second half of the book, as I plan to visit them slightly later in specialized textbooks.
I will now try to summarize what I learned in a few paragraphs. A statement in mathematics is a sentence that is either true or false. Certain statements are taken to be true, these are called the axioms. A mathematical proof of a statement is a sequence of logical deductions leading to the statement from a base set of axioms.
There are different types of proof. In a direct proof, the statement that is being proven is derived directly from the set of assumptions. In a proof by contradiction, the opposite of what is to be proven is assumed to be true and a contradiction is derived. A proof by induction involves two steps: the base step and the inductive step.
Some concepts are common throughout all the subfields of mathematics. Among these are the sets and the functions. A set is a collection of objects. A set may be a subset of another set. There is a unique set with no element called the empty set. Sets may be combined to produce other sets, through operations such as union, intersection and complement.
A function is a mapping from one set (called the domain) to another set (called the codomain). Every element in the domain must map to some element in the codomain, and no element in the domain is mapped to more than one element in the codomain. A function is a special case of a general concept called relations.
In an injective function, no two elements in the domain map to the same element in the codomain. In a surjective function, every element in the codomain is mapped to by some element in the domain. A function that is both injective and surjective is called bijective. Every bijective function has an inverse function that maps elements in the codomain of the original function to those in the domain.
Two sets have the same cardinality (size) if there exists a bijective function from one set to the other. A set is denumerable if it has the same cardinality as the infinite set of positive integers. A set is countable if either it is finite or it is denumerable (also called countably infinite). The set of integers and the set of rational numbers are both countable. The set of real numbers is uncountable. It can be shown that any infinite subset of a denumerable set is denumerable.