Santa Barbara City College Course Outline

MATH 180 - Transition to Advanced Mathematics

MATH 180
Transition to Advanced Mathematics
Disciplines
Mathematics (Masters Required)
4.000
0 - May not be repeated
Designed to introduce students to the rigors of advanced mathematics courses, with an emphasis on reading and writing proofs. Topics include set theory and logic, relations, functions, induction, countable and uncountable sets, the Heine-Borel Theorem and the Bolzano-Weierstrass Theorem. Some elementary group theory and/or topology is covered.
64.000-72.000 Total Hours
Total Hours
128.000-144.000 Total Hours
64.000-72.000 Total Hours
Prerequisite: MATH 160
Prerequisite or Corequisite: None
Concurrent Corequisite: None
Course Advisories: None
Limitation on Enrollment: None
Course Objectives:
Use techniques such as contradiction and contrapositive to prove statements.
Perform set operations using union, intersection, complementation, and DeMorgan's Laws.
Write proofs involving the well-ordering principle and mathematical induction.
Write proofs pertaining to relations, equivalence relations, partitions, and equivalence classes.
Write proofs involving one-to-one functions, onto functions, preimage, and inverse image.
Write proofs pertaining to finite sets, countable sets, and uncountable sets. Apply the Bernstein-Schroder Theorem to prove the equivalence between sets.
Use the Axiom of Choice to prove statements pertaining to the cardinality of a set. Apply the Axiom of Choice to prove the Comparability Theorem.
Use the Heine-Borel Theorem to prove statements involving compact sets.
Use the Bolzano-Weierstrass Theorem to prove statements involving accumulation points and limit points.
Student Learning Outcomes
MATH180 SLO1 - Perform set operations using unions, intersections, complementation, and DeMorgan's laws
MATH180 SLO2 - Prove mathematical facts involving functions, set theory, cardinality, and properties of the real numbers
MATH180 SLO3 - Use techniques such as contradiction, the contrapositive, the well-ordering principle, and mathematical induction in writing proofs.

1.  The Logic of Mathematical Proofs:

a. Propositions and connectives

b. Conditionals and biconditionals

c.  Quantifiers

d.  Mathematical proofs

e.  Proofs involving quantifiers

f.  Induction

2.  Set Theory

a.  Notation and set theory

b.  Set operations

c.  DeMorgan's Laws

3.  Functions

a.  Cartesian Products

b.  Relations

c.  Equivalence relations

d.  Partitions

e.  Functions

f.  Onto functions

g.  One-to-one functions

h.  Induced set functions

3. Cardinality

a.  Equivalent sets

b.  Finite sets

c.  Countable sets

d.  Uncountable sets

e.  Schroder-Bernstein Theorem, the Axiom of Choice, and Zorn's Lemma

4.  The Real Numbers

a.  Properties of real numbers

b.  completeness of real numbers

c.  The Heine-Borel Theorem

d.  The Bolzano-Weierstrass Theorem

Methods of Instruction
Directed Study
Discussion
Lecture
Projects

1.  Prove that any bounded subset of real numbers without an accumulation point is finite.

2.  Let R be an equivalence relation.  Show that xRy if xR=yR

A. Appropriate Readings: Students are required to read assigned sections in text or supplements. B. Writing Assignments: Students must work assigned mathematical problems requiring the manipulation of abstract symbols. C. Appropriate Outside Assignments: Students are expected to spend a sufficient amount of time outside of class to practice techniques presented during class time, read assigned materials, and complete frequent homework assignments. D. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze using both inductive and deductive reasoning within a logical system.
A student's grade will be based on regular assessments: homework, quizzes, and exams. A comprehensive final exam will be given at the end of the semester.
    Transition to Advanced MathematicsSmith, Eggen, St Andre, Brooks Cole, 2014
09/28/2021
CAC Approval: 10/18/2021
Board of Trustees: 10/28/2021