Life of Fred: Five Days of Upper Division Math:

Set Theory

Modern Algebra

Abstract Arithmetic

Topology

Here is a chance to explore experience many of the major parts of upper division math before diving deeply into each of them.

Master teacher, Fred Gauss, presents a week's worth of lectures on each of these topics. And, as usual, makes them much more exciting than 99.99999% of other university teachers can do. In these five days, he will present about a fifth of each of these courses.

He will offer 139 puzzles for readers to solve.

Here's the table of contents:

Monday

Set Theory

cardinality of a set 16, set builder notation 16, union and

intersection 17, subset 17, naive set theory 17, modus ponens 18,

seven possible reasons to give in a math proof 18, the high school

geometry postulates are inconsistent 19, every triangle is isosceles

19, normal sets 22

Modern Algebra

math theories 23, definition of a theorem 24, six properties of

equality 25, binary operations 25, formal definition of a binary

operation 26, formal definition of a function 26, definition of a

group 27, right cancellation law 27, left inverses 27,

commutative law 28

Abstract Arithmetic

circular definitions 30, unary operations 31, successor function S

31, natural numbers 31, the five Peano postulates 32, mathematical

induction 33

Topology

topology is all about friendship 36, listing all possible subsets 37,

open sets 37, the discrete topology 37, the three axioms of a
topology 38, models for a topology 40, open intervals 40

TUESDAY

Set Theory

axiom of extensionality 46, propositional functions 47, Zermelo-

Fraenkel axiom #2 (axiom schema of specification) 48

Modern Algebra

three examples of non-commutative groups 51, uniqueness of right

inverses and right identities 55

Abstract Arithmetic

no number can equal its successor 56, definition of + 57,

recursive definitions 57, proving 2 + 2 = 4 58

Topology

the rationals are dense in the real numbers 62, topology of X when

X is small 65, limit points 65, standard topology for R 66, closed

intervals 66

WEDNESDAY

Set Theory

ZF #3, the axiom of pairing 68, ZF #4, the axiom of union 69

Modern Algebra

(a – 1 ) – 1 = a 71, If a and b are members of a group and if a 2 = e and if

b 2 a = ab 3 , then b 5 = e. 73, defining cardinality in terms of 1–1 onto

functions 75, group isomorphisms 75

Abstract Arithmetic

recursive definition of multiplication in ù 80, proof of the

distributive law 80, definition of nm in ù 81, definition of the least

member of a set in N 83, strong induction 83, total binary relations

84

Topology

derived sets 85, closed sets 85, set subtraction 85, closed sets

THURSDAY

Set Theory

ZF #5, the power set axiom 89, Cartesian products, relations, and

functions 93, domains, codomains, and ranges 93, one-to-one onto

functions and the cardinality of sets 94

Modern Algebra

groups of low order 95, Klein four-group 96, If a and b are

members of a group and if ba 2 = ab 3 and if a 2 b = ba 3 , then a = e.

98, subgroups 99

Abstract Arithmetic

partition of a set 101, equivalence relations 102, equivalence

classes 103, defining the integers as equivalence classes 105,

integer addition 106, integer multiplication 107, well-defined 108,

< in the integers 109, integer subtraction 109, proof that a negative

times a negative gives a positive answer 109

Topology

limit point definition of continuous functions 111, continuous

functions and open sets 113

FRIDAY

Set Theory

ZF #6, axiom of replacement 114, ZF #7, axiom of infinity 116,

inserting all of abstract arithmetic into set theory 117, ZF #8,

axiom of foundation 118, Schröder-Bernstein theorem 119,

inaccessible cardinals and other big cardinals 121,

metamathematics 121

Modern Algebra

cosets 124, cosets are either equal or disjoint 125, Lagrange's

theorem 125, groups, semigroups, monoids, abelian groups, rings,

fields, and vector spaces 126

Abstract Arithmetic

the rational numbers defined as equivalence classes 129, +, ×, –,

and ÷ in Q 130, ways not to define the real numbers 131, cuts in Q

132, real numbers defined 132, most irrational numbers do not

have nice names 134, the complex numbers 135

Topology

separated 136, connected 136 , continuous image of a connected

set is connected 137, open coverings 137, compact 137,

1 2 continuous image of a compact set is compact 138, T1, T2 ,
regular, T3 , normal, and T4 spaces 139

Solutions 140

Index 206

All 208 pages for $19. ← not a typo! This is my gift to all those who are considering becoming math majors.

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