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, œ 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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