The Timpson Lyceum · Real Analysis (2020)

Lecture 1 (Real Analysis 2020)

Overview

This opening lecture lays the logical and set-theoretic groundwork the whole course rests on, then takes the first real step into analysis: the field axioms for the real numbers. The aim is not to "do calculus" but to learn to reason deductively — to take a set of premises and derive their logical consequences. We review the three modes of reasoning (deductive, inductive, abductive) and commit to deductive reasoning alone, then build up propositional logic, sets, relations, and functions. Finally, we treat $\mathbb{R}$ as an undefined set equipped with two undefined operations $+$ and $\cdot$ obeying the field axioms, and prove several familiar arithmetic facts strictly from those axioms.

Key Concepts

Three kinds of reasoning

Deductive — a conclusion that follows with certainty from its premises; the implication is universally valid, independent of any facts about our universe (e.g. "if all men are mortal and Socrates is a man, then Socrates is mortal"). This is the only reasoning permitted in mathematics.
Inductive — generalizing from repeated observations; never a guarantee (you can list evens forever and never prove "every number is even").
Abductive — inferring a particular cause from general patterns (the "best explanation"); used in courts and history, weaker still.

Propositional logic

Statement — a sentence that is either true or false, but not both.
Conjunction $A \wedge B$ (" $A$ and $B$ ") — true only when both are true.
Disjunction $A \vee B$ (" $A$ or $B$ ") — the inclusive or; true unless both are false.
Negation $\neg A$ ("not $A$ ") — the contradiction of $A$ : exactly one of $A,\neg A$ is true. (Contrast contrary statements, which merely cannot both be true.)
De Morgan's laws — negation distributes across a connective and flips it: $\neg(A \wedge B) \equiv \neg A \vee \neg B, \qquad \neg(A \vee B) \equiv \neg A \wedge \neg B.$
Implication $A \Rightarrow B$ ("if $A$ then $B$ ") — false only when $A$ is true and $B$ is false; vacuously true whenever the antecedent $A$ is false. Equivalent to its contrapositive $\neg B \Rightarrow \neg A$ .
Logical equivalence $A \Leftrightarrow B$ — means $A \Rightarrow B$ and $B \Rightarrow A$ ; true exactly when $A$ and $B$ share the same truth value.

Sets

A set is an unordered collection of distinct objects: order and duplicates do not matter, so $\{1,2\}=\{2,1\}=\{1,1,2\}$ .
Membership $x \in A$ ; non-membership $x \notin A$ .
Standard sets — $\varnothing$ (empty), $\mathbb{N}=\{1,2,3,\dots\}$ (naturals start at $1$ here), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals, ratios of integers), $\mathbb{R}$ (reals). These nest: $\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$ .
Set-builder notation — $\{x : P(x)\}$ , read "the set of all $x$ such that $P(x)$ ," e.g. the evens $\{x : x\in\mathbb{Z},\ 2 \mid x\}$ .
Quantifiers — $\forall$ ("for all") and $\exists$ ("there exists").
Subset $A \subseteq B$ iff $\forall x\in A,\ x\in B$ ; proper subset $A \subsetneq B$ iff $A\subseteq B$ and $A \neq B$ . Every set is a subset of itself.
Set equality — $A=B$ iff $A\subseteq B$ and $B\subseteq A$ (mutual inclusion).
Operations — union $A\cup B=\{x : x\in A \vee x\in B\}$ ; intersection $A\cap B=\{x : x\in A \wedge x\in B\}$ ; difference $A\setminus B=\{x : x\in A \wedge x\notin B\}$ . Operations return a set; relations like $\subseteq$ make a statement (true/false).

Lists, relations, functions

A list uses parentheses; order and duplicates do matter — it models a point, e.g. $(1,2,3)$ .
A relation is any set of ordered pairs.
A function is a relation $f$ such that whenever $(a,b)\in f$ and $(a,c)\in f$ , then $b=c$ — each input maps to exactly one output. We write $f(a)=b$ .
Domain $=$ set of all first coordinates; image $=$ set of all second coordinates.
$f:A\to B$ means $\operatorname{dom}(f)=A$ and $\operatorname{image}(f)\subseteq B$ .
Onto (surjective) — image equals $B$ . One-to-one (injective) — $f(x)=f(y)\Rightarrow x=y$ . Bijection — both.

The field axioms for $\mathbb{R}$ We treat $\mathbb{R}$ , $+$ , and $\cdot$ as undefined, assuming only that $+,\cdot$ are closed on $\mathbb{R}$ and satisfy:

Commutativity — $x+y=y+x$ , $\ x\cdot y=y\cdot x$ .
Associativity — $(x+y)+z=x+(y+z)$ , similarly for $\cdot$ .
Additive identity — $\exists\,0$ with $0+x=x$ for all $x$ .
Multiplicative identity — $\exists\,1$ with $1\cdot x=x$ for all $x$ .
Additive inverse — $\forall x,\ \exists y$ with $x+y=0$ .
Multiplicative inverse — $\forall x\neq 0,\ \exists y$ with $x\cdot y=1$ .
Distributivity — $x\cdot(y+z)=x\cdot y + x\cdot z$ .
Nontriviality — $0\neq 1$ (rules out the one-element set).

From these one proves uniqueness of $0$ and of inverses, allowing the definitions $-x$ (the unique additive inverse), $x^{-1}$ (the unique multiplicative inverse, $x\neq0$ ), and $x-y := x+(-y)$ . Substituting equal quantities into an operation ("add the same thing to both sides") is a rule of logic, not an axiom.

Worked Example

Claim. From the field axioms, $0\cdot x = 0$ for every $x\in\mathbb{R}$ .

The key levers are the additive-identity axiom $0+0=0$ , distributivity, and the left cancellation law ( $a+b=a+c \Rightarrow b=c$ , itself proved by adding $-a$ to both sides).

\begin{aligned} 0 &= 0 + 0 &&\text{(additive identity)}\\ x\cdot 0 &= x\cdot(0+0) &&\text{(apply $x\cdot$ to both sides)}\\ x\cdot 0 &= x\cdot 0 + x\cdot 0 &&\text{(distributivity)}\\ x\cdot 0 + 0 &= x\cdot 0 + x\cdot 0 &&\text{(additive identity, LHS)}\\ 0 &= x\cdot 0 &&\text{(left cancellation of $x\cdot 0$)}\\ 0 &= 0\cdot x &&\text{(commutativity)}. \end{aligned}

So no special "multiply-by-zero" axiom is needed — the fact $0\cdot x=0$ is forced by the field axioms alone.

Summary

Mathematics uses deductive reasoning exclusively; inductive and abductive reasoning give likelihood, never proof.
Propositional logic supplies the toolkit: conjunction, disjunction, negation, implication, and equivalence, with De Morgan's laws and the contrapositive as everyday rewrites.
Sets are unordered collections defined up to membership; $\subseteq$ / $=$ are relations (statements), while $\cup$ , $\cap$ , $\setminus$ are operations (returning sets). The standard sets nest $\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$ .
A function is a relation in which each input has exactly one output; key properties are injective, surjective, and bijective.
$\mathbb{R}$ is built axiomatically: $+$ and $\cdot$ are undefined closed operations satisfying the field axioms, from which $0$ , $1$ , $-x$ , and $x^{-1}$ are unique and familiar arithmetic (e.g. $0\cdot x=0$ , cancellation, sign rules) is proved, not assumed.