APPLIED MATHEMATICS

Course Credits: 3 Units

Prerequisites: Math 18

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MATH 107 - Logic and Set Theory

Course Description

MATH 107 is a foundation course that will equip students with the necessary skills in proof construction using axiomatic methods, as guided by the laws in Logic. Likewise, mathematical reasoning can be well established as supported by all the properties from axioms, definitions, corollaries, lemmas, and theorems. The goal of this course is to instill in the students the basic knowledge and skills necessary to create propositions and theorems that need to be proven logically. This course will prepare them to fully understand some mathematical concepts and challenges that await them in the BSAM program.

Course Learning Outcomes

By the end of the semester, the students would be able to:

  1. Discuss the fundamental processes in building set-theoretic concepts.
  2. Identify some basic laws in logic that are evident in proof construction.
  3. Prove mathematical statements using the necessary properties, theorems, and appropriate techniques.
  4. Demonstrate the importance of logic in proving mathematical concepts.
  5. Be equipped with the necessary tools and preparations to learn more about higher mathematics courses.
Course Outline

I. Introduction to Logic

  1. Mathematical Statements
  2. Mathematical Reasoning and the Rules of Inference
  3. Axiomatic Systems and Methods of Proofs

II. Sets and Set Notations

  1. Axioms and Defenitions
  2. Set Relations
  3. Operations on Sets
  4. Proving Properties of Sets
  5. Applications

III. Functions and Relations

  1. Definitions of Functions and Relations
  2. The ordered pair
  3. Domain and Range
  4. Axiom of Choice
  5. Properties of Functions and Relations
  6. Equivalence relation and equivalence classes
  7. Partitions
  8. Partial Orders and Linear Ordering
  9. Hasse Diagram
  10. Inverse Relations
  11. Operation on Relations
  12. Injective and Surjective Maps
  13. The Characteristic Function
  14. The set of functions from A into B

IV. Construction of Real Numbers

  1. Natural Numbers, Definition and Properties
  2. The successor of a set

V. Transfinite Cardinalities

  1. Definitions and Axioms
  2. Finite and Infinite Sets
  3. The Aleph Null/Nought
  4. The Power of the Continuoum
  5. Countable and Denumerable Sets
  6. Uncountable and Non-Denumerable Sets
  7. Equipotence and Equipotence to a subset
  8. Geometric and Analytic Proof of Equipotence
  9. The Schoeder Bernstein Theorem
  10. The Cantor Theorem
  11. Integers