Rather than a course of study, this book is a sampler of
advanced math for students who love math and Fred and want to
preview courses they may take later.

Upper division (college junior/senior) pure math is much
different than calculus. No “word problems,” no formulas to
memorize, no concrete applications—just puzzles to solve. Instead
of learning procedures, students create definitions, theorems, and
proofs.

These are the first five days of Fred’s teaching set theory,
modern algebra, abstract arithmetic, and topology. Each of the 139
assignments/puzzles/questions that he gives his students calls for
creativity rather than doing drill work. Some of these can be done
in a minute. Some will take several hours to complete. They are all
meant to be enjoyed.

The first day of set theory- cardinality of a set, set builder
notation, naive set theory, modus ponens, seven possible reasons to
give in a math proof, the high school geometry postulates are
inconsistent, the proof that every triangle is isosceles, normal
sets.

The first day of modern algebra- definition of a math theory,
six properties of equality, formal definition of a binary
operation, formal definition of a function, definition of a group,
right cancellation law, left inverses, commutative law.

The first day of abstract arithmetic- circular definitions,
unary operations, the successor function, natural numbers, the five
Peano postulates, mathematical induction.

The first day of topology- topology is all about friendship,
listed and counting subsets, open sets, the discrete topology, the
three axioms of a topology, models for a topology, open
intervals.

By the fifth day Fred will have covered the Schröder-Bernstein
theorem (set theory), proved Lagrange’s theorem for subgroups of
any group (modern algebra), defined the real numbers based only on
the concept of “adding one” (abstract arithmetic), and explored
continuous images of compact sets (topology).