9/1/2024 3:21:40 AM |
| New Course (First Version) |
CATALOG INFORMATION
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Discipline and Nbr:
MATH 6 | Title:
INTRO TO HIGHER MATH |
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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:
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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:
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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: | | |
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CSU GE: | Transfer Area | | Effective: | Inactive: |
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IGETC: | Transfer Area | | Effective: | Inactive: |
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CSU Transfer: | Transferable | Effective: | Spring 2013 | Inactive: | |
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UC Transfer: | Transferable | Effective: | Spring 2013 | Inactive: | |
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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:
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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
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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:
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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. |
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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:
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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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