How to Think Like a Mathematician: a Logbook

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This is my second attempt to read this book after failing to do so last year. I wish that, after applying the PACER method from Dr Justin Sung, I could learn from this book with more discipline.

August 13th, 2026

First session

Topic: Sets and Functions

What I learnt:

  • Definition of a set and a function
  • Sets of numbers: natural, integer, non-negative integer, rational, real, and complex
  • Definition of an empty set, cardinality, subset, and proper subset
  • Set builder notation
  • Operation on sets
  • Domain and range

Insight:

  • At first, I didn’t understand why this book begins with sets and functions as its first chapter. I finally understand why this topic matters because “most high-level mathematics is about sets and functions between them,” as I quote from the book. Or basically every mathematician’s building block.
  • In this chapter, I learnt that Calculus is actually the study of functions from the set of real numbers to the set of real numbers that have the property that we can differentiate them.
  • “For any set X, the subset X is not a proper subset of X.” I was stuck here for maybe 5 minutes. After careful reading, I noticed that subset X is actually the set X itself. This is my aha moment.
  • The symbol for subset ⊆ (inclusive) is actually <=. Same with proper subset ⊂, which is actually < (not inclusive, as if X is a proper subset of Y, then at least one element of X is not an element of Y.
  • I got 100 marks on all of the exercises!

Next: Chapter 2 Reading Mathematics

Second session

Topic: Reading mathematics

What I learnt: How to read a mathematical textbook and tips

Insight:

  • The key to learning mathematics is actually to be active! Never read it like a novel. Always use pen and paper.

  • We can use the first chapter of high-level books as a summary, since most of them provide a good summary of the basics

  • Tips from the book:

    1. Skim the chapter first. Note something that is important (appears multiple times throughout the section)
    2. Ask questions
    3. Read through carefully. Always start with the statement; the proof later, if we like.
    4. Be active. This should include checking the text and doing the exercises. Never trust the author!
    5. Always reflect

Next: Writing mathematics I, II

August 14th, 2026

Topic: Writing mathematics I, II

What I learnt:

  • Rules of writing mathematics

Insight:

  • Always write in sentences with proper grammar (s + v), use punctuation, and keep it simple; use simple sentences. Break it down if it’s too long.
  • Always define your symbols and notation first!

Next: How to solve a problem