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MATH
211 Introduction to Mathematical Proof (3 hrs)
A. Catalog description:
A transition to upper level mathematics courses. This course highlights various
types of mathematical thinking including direct, indirect and inductive proofs,
with careful treatment of quantified statements. Topics include sets, number
theory and relations. Development of the ability to write a precise mathematical
proof is the primary goal.
B. Prerequisite: MATH 121 and MATH 122.
C. Course objectives:
The primary objective of this course is to develop a certain amount of fluency
with the language of mathematics and basic techniques of mathematical proof.
Students should understand the significance of a quantified statement –
what it means if such a statement is true, and what it takes to show that
such a statement is false. The emphasis should be on the writing of proofs.
We want the student to learn the rules of logic, develop a good writing style,
explore some interesting mathematics and learn to make conjectures, decide
if the conjecture is true or false, and write up a justification (proof or
counterexample). It is intended that a student completing this course will
have had success with proving and disproving mathematical statements. As a
consequence, the emphasis should be on the collecting and recollecting of
attempts at proof, until the student has some success. It is hoped that this
success will be sufficient to enable the student to cope with rigorous “proof-oriented”
courses such as MATH 411 Abstract Algebra, MATH 416 Linear Algebra and MATH
420 Introduction to Analysis. This course is not intended for students who
have already successfully completed 400 level courses such as those mentioned
above.
D. Usual course content:
The topics that should be covered include mathematical induction, the basics
of set theory, equivalence relations, cardinality (including the notions of
countability and uncountability), and methods of proof. Other topics should
probably include an introduction to symbolic logic, which stresses logical
implication, quantifiers, contrapositive, logical equivalence, etc. (but does
NOT stress symbolic manipulation or truth tables) and some coverage of functions
– surjective, injective, bijective, and inverse functions. Some topics
that might be included or touched upon include counting and probability, algebraic
structures – binary operations and properties, the Cantor/Schroeder/Bernstein
Theorem, and the binomial theorem. An instructor may introduce any topic that
contributes to the goals of the course
E. Technology: None.
F. Students who may benefit from the course:
This is a prerequisite for all of the upper level courses including geometry,
abstract and linear algebra and complex and real analysis. Students in the
secondary education curriculum must take this course. Nonteaching mathematics
majors must take this course. This course could prove valuable for students
who are interested in following careers in theoretical physics and/or theoretical
computer science.
G. Follow up courses:
MATH 341 College Geometry, MATH 411 Abstract Algebra, MATH 416 Linear Algebra,
MATH 420 Introduction to Analysis, MATH 424 Complex Analysis.
H. Textbooks used in the past:
1. Chapter Zero, Carol Schumacher, Addison-Wesley, (ISBN 0-201-43724-4).
2. Foundations of Higher Mathematics: Exploration and Proof, Fendel and Resek,
Addison-Wesley, (ISBN 0-201-12582-0).
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