View source for Introduction to abstract mathematics: course

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==Description==

This is a one-semester course to introduce math majors into pure/abstract mathematics. It's mostly about ''proofs''. However, I would like to introduce a few other things that will make [[linear algebra]], [[advanced calculus: course|advanced calculus]], [[Group theory: course|group theory]], [[topology]], etc easier to absorb later on.

Aka:
*Introduction to higher mathematics
*Introduction to advanced mathematics
*Introduction to proofs

==Prerequisites== 
*[[Introductory calculus: course|Calculus 1-3]]


==Outline==

About proofs... 

"Because I don’t have time to write you a short letter, I write you a long one." Attributed to Hemingway, Cicero, Voltaire, Mark Twain, Blaise Pascal, Goethe.

In writing proof we'd better follow the latter approach, especially when we are still learning.

"If you like laws and sausages, you should never watch either one being made." (Mis)attributed to Bismarck.

We'll ''have'' to watch, if we want to learn how to do it.

In addition to proofs we need to start to become familiar with various mathematical constructions. They should be illustrated with examples from previous courses ([[calculus]] and precalculus).

*[[Rules of proof writing]]
*Introduction: 
**Sloppy notation.
**examples and types of proofs: 
***computation,
***contradiction, 
***induction, 
***counterexamples.
*Proofs:
**reading proofs,
**checking proofs,
**discovering proofs,
**writing proofs.
*Trivialization as a way of understanding concepts, proofs etc
*Examples:
**[[Proofs in Euclidean geometry]]
**[[Proofs about numbers]]
**[[Proofs in calculus]]
*Abstract mathematics, axioms, definitions, and well-defined concepts: 
**turning theorems into definitions,
**axioms of Euclidean geometry,
**axioms of topology,
**coordinates,
**[[limit]], 
**[[Riemann integral]], 
**[[PageRank]], 
**circumnavigation
**definition-theorem-proof.
*Sets: 
**[[subset]]s, 
**the [[power set]], 
**[[union]]s, [[intersection]]s, [[complement]]s 
*Functions: 
**[[identity function]], [[constant function]]s, 
**[[composition]]s, 
**[[commutative diagram]]s:
***algebra of limits,
**[[one-to-one]] and [[onto]], 
**[[bijection]]s, [[inverse]]s, 
**[[image of function|image]]s and [[preimage]]s, 
**[[inclusion]]s, [[restriction]]s, [[extension]]s.
*New sets from old: 
**[[product set]]s and [[projection]]s
**[[equivalence relation]]s, [[partition]]s, 
**[[quotient sets]], the [[quotient function]]:
***[[antiderivative]]s, 
***[[modular arithmetic]]
*New sets and functions from old: 
**function-valued functions, 
**spaces of functions, 
**dual spaces, duality

==Review==
*[[Is a set a subset of itself? ]]
*[[Is projection a quotient function? ]]


==Notes==
Text:

==Further reading==
*[[Linear algebra: course]]
*[[Group theory: course]]


[[Category:Mathematics]] [[Category:Sets]] [[category:courses]] [[category:sets]]