Class notes
Mam2000w - Real Analysis cheat sheet
- Course
- MAM2000W
- Institution
- University Of Cape Town (UCT)
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Harold’s Real Analysis Cheat Sheet22 October 2022
Number Sets
Symbol Definition Examples Equations Solution
∅ 1=2
empty set,
{} null
ℕ1 = {1, 2, 3, …}
set with no members
ℕ
ℕ0 = {0, 1, 2, 3, …}
Pre-2010 NA
natural numbers
ℙ
See ISO 80000-2 2-6.1
ℤ 𝑥+7=0 𝑥 = −7
prime numbers {2, 3, 5, 7, 11, 13, ...} unofficial NA
ℚ 4𝑥 − 1 = 0 𝑥=¼
integers {…, −2, −1, 0, 1, 2, …}
𝔸 {5, -7, ½, √2} 2𝑥 2 + 4𝑥 − 7 = 0 x is algebraic
rational numbers {0, ¼, ½, ¾, 1}
algebraic numbers
𝕋 𝕋=𝕌−𝔸
transcendental
{π, e, eπ, sin(x), logb a} NA
ℝ {3.1415, -1, ⅞, √2, π} 𝑥2 − 2 = 0 𝑥 = ±√2
numbers
𝑥 = ±√−1
real numbers
𝕀 {2i, √−1} 𝑥2 + 1 = 0
𝑥 = ±𝑖
imaginary numbers
ℂ 𝑥 2 − 4𝑥 + 5 = 0 𝑥 =2±𝑖
𝕌
complex numbers {1 + 2i, -3.4i, ⅝}
universal set {all possible values} ∞ NA
Derived Number Sets
Copyright © 2021 by Harold Toomey, WyzAnt Tutor 1
, Integers ℤ
Symbol Definition Equations Examples
ℤ*
{0} zero n=0 {0}
ℤ - {0}
ℤ \ {0}
non-zero integers n≠0 {-3, -2, -1, 1, 2, 3, …}
ℤ+
ℕ ⋃ {0}
positive integers n>0 {1, 2, 3, …}
ℤ‒
non-negative integers n≥0 {0, 1, 2, 3, …}
ℤ‒ ⋃ {0}
negative integers n<0 {…, -3, -2, -1}
Real Numbers ℝ
non-positive integers n≤0 {…, -3, -2, -1, 0}
ℝ - {0}
{0} zero x=0 {0.0}
ℝ \ {0}
non-zero real numbers x≠0 {-0.001, 0.001}
ℝ+
positive real numbers x>0 {0.0001, 0.0002, ...}
ℝ+ ⋃ {0}
(0, ∞)
non-negative real numbers x≥0 {0, 0.0001, 0.0002, ...}
ℝ‒
[0, ∞)
negative real numbers x<0 {…, -0.0002, -0.0001}
ℝ‒ ⋃ {0}
(-∞, 0)
non-positive real numbers x≤0 {…, -0.0002, -0.0001, 0}
(-∞, 0]
Copyright © 2021-2022 by Harold Toomey, WyzAnt Tutor 2
,Definitions
Term Definition
A precise and unambiguous description of the meaning of a mathematical term. It
Definition characterizes the meaning of a word by giving all the properties and only those
properties that must be true.
A mathematical statement that is proved using rigorous mathematical reasoning.
Theorem In a mathematical paper, the term theorem is often reserved for the most
important results.
A minor result whose sole purpose is to help in proving a theorem. It is a
steppingstone on the path to proving a theorem. Very occasionally lemmas can
Lemma
take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s
lemma, Sperner’s lemma).
A result in which the (usually short) proof relies heavily on a given theorem (we
Corollary
often say that “this is a corollary of Theorem A”).
Proposition A proved and often interesting result, but generally less important than a theorem.
A statement that is unproved, but is believed to be true (Collatz
Conjecture
conjecture, Goldbach conjecture, twin prime conjecture).
Claim An assertion that is then proved. It is often used like an informal lemma.
A statement that is assumed to be true without proof. These are the basic building
Axiom /
blocks from which all theorems are proved (Euclid’s five postulates, Zermelo-
Postulate
Fraenkel axioms, Peano axioms).
A mathematical expression giving the equality of two (often variable) quantities
Identity
(trigonometric identities, Euler’s identity).
A statement that can be shown, using a given set of axioms and definitions, to be
both true and false. Paradoxes are often used to show the inconsistencies in a
Paradox flawed theory (Russell’s paradox). The term paradox is often used informally to
describe a surprising or counterintuitive result that follows from a given set of
rules (Banach-Tarski paradox, Alabama paradox, Gabriel’s horn).
Corollary
↑
Theorem /
Proposition
↑
Lemma
↑
Axiom / Postulate
↑
Conjecture / Claim
↑
Definition
Textbook Bloch, Ethan D.. The Real Numbers and Real Analysis. Springer New York, 2011.
Copyright © 2021-2022 by Harold Toomey, WyzAnt Tutor 3
, Ch. 1.2: Natural Numbers ℕ
Axiom / Theorem /
Description
Lemma / Definition
Let S be a set.
Operations: Binary, Unary
A binary operation on S is a function S × S → S.
(Definition 1.1.1)
There exists a set ℕ with an element 1 ∈ ℕ and a function s: ℕ →
A unary operation on S is a function S → S.
ℕ that satisfy the following three properties.
Peano Postulates a. There is no n ∈ ℕ such that s(n) = 1.
c. Let G ⊆ ℕ be a set. Suppose that 1 ∈ G, and that if g ∈ G
(Axiom 1.2.1) b. The function s is injective.
then s(g) ∈ G. Then G = ℕ.
Natural Number The set of natural numbers, denoted ℕ, is the set the existence of
Let a ∈ ℕ. Suppose that a ≠ 1.
(Definition 1.2.2) which is given in the Peano Postulates.
Then there is a unique b ∈ ℕ such that a = s(b).
Lemma 1.2.3
Let H be a set, let e ∈ H and let k: H → H be a function. Then there
is a unique function f: ℕ → H such that f(1) = e, and that f ◦ s = k ◦ f.
Definition by Recursion
There is a unique binary operation +: ℕ × ℕ → ℕ that satisfies the
(Theorem 1.2.4)
Operation: + following two properties for all n,m ∈ ℕ.
(Theorem 1.2.5) a. n + 1 = s(n). (successor).
There is a unique binary operation *: ℕ × ℕ → ℕ that satisfies the
b. n + s(m) = s(n + m). [= n + (m+1)]
Operation: * following two properties for all n,m ∈ ℕ.
(Theorem 1.2.6) a. n * 1 = n.
Let a, b, c ∈ ℕ.
b. n * s(m) = n(m+1) = (n * m) + n.
1. If a + c = b + c, then a = b (Cancellation Law for Addition).
2. (a + b) + c = a + (b + c) (Associative Law for Addition).
Addition Laws
3. 1 + a = s(a) = a + 1.
(Theorem 1.2.7a)
4. a + b = b + a (Commutative Law for Addition).
5. a + b ≠ 1.
Let a, b, c ∈ ℕ.
6. a + b ≠ a.
7. a * 1 = a = 1 * a (Identity Law for Multiplication).
8. (a + b)c = ac + bc (Distributive Law).
Multiplication Laws 9. ab = ba (Commutative Law for Multiplication).
(Theorem 1.2.7b) 10. c(a + b) = ca + cb (Distributive Law).
11. (ab)c = a(bc) (Associative Law for Multiplication).
12. If ac = bc then a = b (Cancellation Law for Multiplication).
The relation < on ℕ is defined by a < b if and only if there is some p
13. ab = 1 if and only if a = 1 = b.
∈ N such that a + p = b, for all a,b ∈ N.
Relation: <
The relation ≤ on ℕ is defined by a ≤ b if and only if a < b or a = b,
(Definition 1.2.8a)
for all a,b ∈ ℕ.
Relation: ≤
(Definition 1.2.8b)
Copyright © 2021-2022 by Harold Toomey, WyzAnt Tutor 4
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