Mathematical Reasoning – Introduction to Higher Mathematics
LOGIC
Propositional Logic: http://www.cs.ox.ac.uk/people/michael.wooldridge/teaching/soft-eng/lect07.pdf
Propositional Logic and Proofs: http://math.berkeley.edu/~hutching/teach/proofs.pdf
Propositions and Arguments: http://assets.press.princeton.edu/chapters/s9727.pdf
Logic and Argumentation: https://courses.umass.edu/phil110-gmh/text/c01.pdf
CategoricalArguments: https://www.stetson.edu/artsci/philosophy/media/G.%20%20Chapter%206.pdf
Square of Opposition: https://plato.stanford.edu/entries/square/
Syllogistic Reasoning: http://www.logicinaction.org/docs/ch3.pdf
Arguments in Ordinary Language: https://www.sfu.ca/~jeffpell/Phil467/HensonOrdLang67.pdf
METHODS of PROOF
Proofs and Mathematical Reasoning: https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf
Methods of Proofs: http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s3_1.pdf
Direct Proof: http://faculty.up.edu/wootton/Discrete/Section3.1.pdf
Contrapositive Proof: https://math.dartmouth.edu/~m22x17/misc/LaLonde2012_proof_by_contrapositive.pdf
Proof by Contradiction: http://www.math.stonybrook.edu/~brunyate/notes3.pdf
SET THEORY and CARDINALITY
Basic Set Theory: https://www.bu.edu/linguistics/UG/course/lx502/_docs/lx502-basic%20set%20theory.pdf
Alternative Axiomatic Set Theories: https://plato.stanford.edu/entries/settheory-alternative/
Infinite Sets: https://people.umass.edu/gmhwww/382/pdf/09-infinite%20sizes.pdf
Countable and Uncountable Sets: https://www.math.brown.edu/reschwar/MFS/handout8.pdf
Cardinality: https://www.people.vcu.edu/~rhammack/BookOfProof2/Cardinality.pdf
DEDUCTIVE REASONIING and AXIOMATICS METHOD
Deductive Reasoning: http://matt.colorado.edu/teaching/highcog/fall8/j99.pdf
Axiomatic Method: http://www.ms.uky.edu/~droyster/courses/fall11/MA341/Classnotes/Lectures%202&3.pdf
Axiomatic Systems: https://people.math.harvard.edu/~mazur/papers/Axiomatic.pdf
Euclidean Geometry: https://sites.math.washington.edu/~lee/Books/AG/amstext-21-prev.pdf
Non-Euclidean Geometry: https://sites.math.washington.edu/~king/coursedir/m445w05/notes/non-euclidean.pdf
WHAT IS ABSTRACT ALGEBRA?
Introduction to Abstract Algebra: https://math.berkeley.edu/~apaulin/AbstractAlgebra.pdf
Algebraic Structures: http://www.maths.qmul.ac.uk/~pjc/notes/algstr.pdf
Algebraic Group: http://shell.cas.usf.edu/~wclark/Elem_abs_alg.pdf
Algebraic Groups, Rings, and Fields: https://people.maths.ox.ac.uk/flynn/genus2/sheets0405/grfnotes1011.pdf