Mathematics, Statistics & Computer Science
University of Wisconsin-Stout
Jarvis Hall Science Wing 231
Menomonie, WI 54751-0790
COURSE NUMBER/TITLE:
MATH-180 FOUNDATIONS OF THE LANGUAGE OF MATHEMATICS
CREDITS:
4
COURSE DESCRIPTION:
Conceptual foundations of mathematics and the language of mathematics.
Logical rules of inference, statement forms, methods of proof, universal and
existential quantifiers, abstraction and definition, set theory, relations,
functions and cardinality. Proofs involving the above concepts, and
applications of the concepts and methods of proof to particular abstract
mathematical objects. Prerequisite: MATH-151 Calculus with Precalculus
B, MATH-153 Calculus I, or MATH-156 Calculus & Analytic Geometry I.
TEXTBOOK:
A Transition to Advanced Math, 7th Ed., by Smith (adopted Spring 2013)
(6th Ed. adopted S10)
COURSE OBJECTIVES:
Upon completion of this course, the student shall be able to:
1. Work effectively with the logical rules of inference, logical connectives, negation and truth tables in
determining the truth values of logical statements.
2. Work effectively with universal and existential quantifiers used in compound statements and
determine the truth values of such statements.
3. Understand the roles of example and counterexample in mathematical proofs, and use example and
counterexample appropriately in the context of proof.
4. Apply mathematical definitions in the context of examples and proof.
5. Identify the basic methods of proof including direct proof, proof by contradiction, proof by induction
and proof by cases, and be able to apply these methods in the construction of new proofs.
6. Understand the basic concepts of set theory and use the concepts in proofs and examples. These
concepts include the equality of sets, set inclusion, union, intersection, complement, the Cartesian
product and the power set.
7. Work proficiently with the formal definitions of relation, function and equivalence relation in
examples and in the creation of formal proofs and counterexamples.
8. Work proficiently in proofs and examples with concepts involving functions including injectivity,
surjectivity, bijectivity, composition, the inverse of a function, the inverse image of a set under a
function and the image of a set under a function. Combine these concepts and the set theory concepts
in the creation of proofs and counterexamples.
9. Demonstrate a thorough understanding of cardinality and counting by identifying finite, countable
and uncountable sets and proving the sets to be such.
10. Apply the knowledge and skills from objectives 1-9 in the study of the basics of a subject field of
mathematics, probability and statistics or computer science such as number theory, cryptography or
abstract algebraic structures.
COURSE OUTLINE:
I. Statements and Proofs
A. Statements, definitions & axioms (Objective 4)
B. Propositions and theorems (Objectives 1, 4)
C.
D.
E.
F.
Logical connectives and truth tables (Objective 1)
Universal and existential quantifiers (Objective 2)
Conditional and biconditional statements (Objectives 1, 2, 5)
Construction of new statements from old (Objectives 1, 2, 5)
a. Converse
b. Contrapositive
c. Negation
G. Logical equivalence (Objectives 1, 2, 3, 5)
H. Tautologies, contradictions (Objectives 1, 5)
I. Proofs (Objectives 1, 2, 3, 4, 5)
a. Rules of inference
b. Direct proof
c. Proof by contradiction
d. Proof by cases
e. Contrapositive conditional proof
f. Proof by counterexample
g. Proof by induction
h. Equations in proof
II. Set Theory
A. Description of sets (Objective 6)
B. Familiar examples of sets such as integer, rational, real and complex number sets (Objective 6)
C. Set notation and definition (Objective 6)
a. Membership
b. Subset
c. Equality
d. Universal set
e. The empty set
D. Algebra of sets and related proofs (Objectives 1, 2, 3, 4, 5, 6)
a. Union
b. Intersection
c. Complement
d. Power set
e. Cartesian product
f. General Cartesian products
g. Indexed collections of sets
III. Relations and Functions
A. Definitions and related proofs for relations (Objectives 1, 2, 3, 4, 5, 6, 7)
a. Reflexive, symmetric and transitive relations
b. Equivalence relations
c. Partitions
d. Quotient set
e. Composition, union and intersection of relations
f. Inverse relations
B. Definitions and related proofs for functions (Objectives 1, 2, 3, 4, 5, 6, 7, 8)
a. Domain, codomain
b. Well-defined functions
c. Injective, surjective and bijective functions
d. Pigeon hole principle
e. Inclusion, restriction maps on subsets, projection maps on Cartesian products
f. Composition
g. Inverse functions
h. Kernel, quotient set, and the associated commuting diagram
i.
Sets induced by functions
1) Image of a set
2) Inverse image of a set
C. Cardinality of sets: definitions, related proofs and examples (Objectives 1, 2, 3, 4, 5, 6, 7, 8, 9)
a. Cardinality of common sets such as the integer, rational, real, complex number sets
b. Finite, infinite sets
c. Countable sets, denumerable sets, uncountable sets
d. Order relations and ordinal numbers
e. Union of sets
f. General Cartesian product of sets
g. The set of algebraic numbers
h. The set of transcendental numbers
i. Cantor’s theorem
IV. Selected Topics (Objectives 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
A. Number Theory
B. Real number system
C. Binary operations
D. Other survey topics
Updated 9/2015
Revised 3/2013
10/2008