4/18/2024 7:59:09 PM	MATH 6 Course Outline as of Spring 2013	New Course (First Version)
CATALOG INFORMATION
Discipline and Nbr:  MATH 6	Title:  INTRO TO HIGHER MATH
Full Title:  An Introduction to Higher Mathematics
Last Reviewed:12/12/2023

Units		Course Hours per Week			Nbr of Weeks	Course Hours Total
Maximum	4.00	Lecture Scheduled	4.00		17.5 max.	Lecture Scheduled	70.00
Minimum	4.00	Lab Scheduled	0		6 min.	Lab Scheduled	0
		Contact DHR	0			Contact DHR	0
		Contact Total	4.00			Contact Total	70.00
		Non-contact DHR	0			Non-contact DHR Total	0

Total Out of Class Hours:  140.00

Total Student Learning Hours: 210.00

Title 5 Category:  AA Degree Applicable
Grading:  Grade Only
Repeatability:  00 - Two Repeats if Grade was D, F, NC, or NP
Also Listed As: 
Formerly: 

Catalog Description:

A lower division course introducing topics in higher mathematics including introductory set theory and formal logic, proof techniques, mathematical induction, equivalence relations, functions and cardinalities of sets as applied to number theory, calculus and modern algebra.

Prerequisites/Corequisites:
Course Completion of MATH 1B

Recommended Preparation:

Limits on Enrollment:

Schedule of Classes Information
Description:

A lower division course introducing topics in higher mathematics including introductory set theory and formal logic, proof techniques, mathematical induction, equivalence relations, functions and cardinalities of sets as applied to number theory, calculus and modern algebra.

(Grade Only)
Prerequisites:Course Completion of MATH 1B
Recommended:
Limits on Enrollment:
Transfer Credit:CSU;UC.
Repeatability:00 - Two Repeats if Grade was D, F, NC, or NP

ARTICULATION, MAJOR, and CERTIFICATION INFORMATION

Associate Degree:		Effective:		Inactive:
Area:
CSU GE:	Transfer Area			Effective:	Inactive:
IGETC:	Transfer Area			Effective:	Inactive:
CSU Transfer:	Transferable	Effective:	Spring 2013	Inactive:
UC Transfer:	Transferable	Effective:	Spring 2013	Inactive:
C-ID:

Certificate/Major Applicable: Not Certificate/Major Applicable

COURSE CONTENT

Outcomes and Objectives:
At the conclusion of this course, the student should be able to:

Upon successful completion of the course, students will be able to:

1.   Use set theory and logic to convey and understand mathematical concepts.

2.   Prove and disprove mathematical conjectures.

3.   Provide proper counterexamples.

4.   Prove mathematical theorems using mathematical induction and strong mathematical induction.

5.   Characterize functions as injective, surjective, bijective or otherwise.

6.   Characterize relations in terms of the reflexive, symmetric and transitive properties.

7.   Determine when a relation is an equivalence relation.

8.   Characterize a set in terms of its cardinality.

9.   Apply techniques of proofs to number theory, calculus, and topics in modern algebra.

Topics and Scope

I.      Logic

       A. Negation, conjunction and disjunction

       B. Logical form and equivalence

       C. Conditional statements

       D. Biconditional statements

       E. Valid and invalid arguments

       F. Predicates

       G. Quantified statements

       H. Arguments with quantified statements

II.     Set Theory

       A. Introduction to sets

       B. Algebra of sets

       C. Venn diagrams and conjectures

       D. Arbitrary unions and intersections

III.    Proofs

       A. Trivial and vacuous proofs

       B. Direct proofs

       C. Indirect proofs

           i. Proof by contradiction

           ii. Proof by contrapositive

        D. Existence proofs

        E. Counterexamples

        F. Proofs by cases

IV.    Mathematical Induction

       A. Introduction to mathematical induction

       B. Proof by minimum counterexample

       C. Strong mathematical induction

V.     Functions

       A. Definition

       B. Injective and surjective functions

       C. Bijective functions

       D. Composition of functions

       E. Inverse functions

       F. Permutations

VI .   Relations

       A. Relations on sets

       B. Properties of relations

       C. Equivalence relations

       D. Equivalence classes

       E. Congruence modulo n

       F.  Integers modulo n

VII.  Cardinalities of Sets

       A. Numerically equivalent sets

        B. Denumberable sets

       C. Uncountable sets

       D. Comparison of set cardinalities

VIII. Topics in Number Theory

       A. Divisibility properties of integers

       B. The division algorithm

       C. Greatest common divisors

       D. The Euclidean algorithm

       E. Relatively prime integers

       F. The Fundamental Theorem of Arithmetic

       G. Concepts involving sums of divisors

IX.    Topics in Calculus

       A. Limits of sequences

       B. Infinite series

       C. Limits of functions and fundamental properties

       D. Continuity

       E. Differentiability

X.     Various topics from modern algebra that could include

       A. Binary operations

       B. Groups

       C. Rings

       D. Fields

       E. Integral domains

       F. Fundamental properties

       G. Isomorphism

       H. Homomorphisms

       I. Subgroups

Assignments:

1. Daily reading outside of class (approximately 10-50 pages per week).

2. Homework assignments (15-30) consisting of 5-35 problems from required text(s) or supplementary

   materials chosen by the instructor.

3. Quizzes (0-8) and exams (2-6), including final exam.

4. Projects (0-3): research papers on a specific topic (5 to 10 pages)

    or presentations given as posters or short talks. Papers and presentations

    must be related to topics taught in the course.

Methods of Evaluation/Basis of Grade.

Writing: Assessment tools that demonstrate writing skill and/or require students to select, organize and explain ideas in writing.		Writing 0 - 0%
None
This is a degree applicable course but assessment tools based on writing are not included because problem solving assessments are more appropriate for this course.
Problem solving: Assessment tools, other than exams, that demonstrate competence in computational or non-computational problem solving skills.		Problem Solving 5 - 20%
Homework problems
Skill Demonstrations: All skill-based and physical demonstrations used for assessment purposes including skill performance exams.		Skill Demonstrations 0 - 0%
None
Exams: All forms of formal testing, other than skill performance exams.		Exams 70 - 95%
Quizzes; exams, including final exam: problem solving and objective questions
Other: Includes any assessment tools that do not logically fit into the above categories.		Other Category 0 - 10%
Projects: research papers or presentations

Representative Textbooks and Materials:

Mathematical Proof: A Transition to Advanced Mathematics 2/E. Chartrand, Polimeni and Zhang. Pearson: 2008

Introduction to Advanced Mathematics 2/E. Barnier and Feldman. Pearson: 2000 (classic)

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