Introduction to Analysis · Chapter 01

Sets, functions and mathematical language

The vocabulary the rest of the course is written in: membership, inclusion, the operations and their laws, families indexed by a set, and then functions — images, preimages, and what it means for two sets to have the same size.

f : X \to Y

A \cup B

A \cap B

f^{- 1} (U)

A \times B

A \subseteq B

N

x \in A

Lessons

Build the foundation

06

Products, relations and functionsEverything after this chapter is a statement about functions, so the word has to be pinned down. A function is not a formula: it is a rule that assigns to each input exactly one output, together with the two sets it runs between. Two rules that agree everywhere are the same function even when their formulas look nothing alike, and one rule can fail to be a function for a reason that has nothing to do with arithmetic — it may leave an input unanswered, or answer it twice.

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07Images and preimagesA function sends points to points, but it also sends whole sets to whole sets, and it does so in two directions. Pushing a set forward gives its image: everything the function hits from inside that set. Pulling a set back gives its preimage: everything that lands inside it. The two look symmetric and are not, and the difference decides which of them the rest of analysis is built on.Read lesson →

08Injective, surjective and bijective mapsTwo things can go wrong when you try to run a function backwards. It may have sent two inputs to one output, so the backward rule cannot choose; or it may have missed part of the codomain, so the backward rule has nothing to say there. Ruling out the first is injectivity, ruling out the second is surjectivity, and a function with neither fault can be reversed exactly.Read lesson →

09Countable and uncountable setsTwo finite sets have the same size when their elements can be matched up one for one. Nothing in that sentence mentions counting, so it can be asked of infinite sets too — and the answer is not what intuition expects. The whole numbers, the even numbers and the fractions can all be matched up with each other, so in this sense they are the same size. The real numbers cannot be matched with any of them, and that gap is the first genuinely surprising fact in the course.Read lesson →

10Chapter practiceBring the chapter together: read membership statements, compare sets, and compute operations. These questions combine familiar ideas without a worked example immediately above the answer choices.Read lesson →

Mapping the lesson.

Sets, functions and mathematical language

Build the foundation