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Dec 21, 2024
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MA 1830 - Transition to Advanced Topics in Mathematics4 lecture hours 0 lab hours 4 credits Course Description Introduction to proof techniques to be used in upper-level mathematics courses. Topics include logic and proofs, set theory, relations and partitions, functions, and cardinality of sets. (prereq: only open to AS students) Course Learning Outcomes Upon successful completion of this course, the student will be able to:
- Demonstrate proficiency in elementary logic, including using truth tables to prove logical equivalence
- Manipulate logical sentences symbolically and semantically: for example, apply DeMorgan’s Law to construct denials
- Demonstrate familiarity with the natural numbers, integers, rational numbers, real numbers, and complex numbers
- Demonstrate proficiency in interpreting and manipulating existential and universal quantifiers
- Read and construct proofs using direct and indirect methods
- Choose methods of proof appropriately
- Read and construct proofs involving quantifiers
- Demonstrate proficiency in elementary set theory including construction of sets, subsets, power sets, complements, unions, intersections, and Cartesian products
- Interpret unions and intersections of indexed families of sets
- Read and construct proofs involving set theoretic concepts
- Apply the principle of mathematical induction and its equivalent forms
- Manipulate summations in sigma notation
- Read and construct proofs related to relations, equivalence relations, and partitions of sets
- Demonstrate familiarity with functions as relations; injections, surjections, and bijections
- Construct functions from other functions: for example, compositions, restrictions, and extensions
- Read and construct proofs related to functions
- Demonstrate familiarity with cardinality for finite, countable, and uncountable sets
Prerequisites by Topic Course Topics
- Elementary logic with truth tables
- Quantifiers
- Methods of proof
- Elementary set theory
- Operations with sets including indexed families of sets
- Principle of mathematical induction and its equivalent forms
- Cartesian products
- Relations, equivalence relations, and partitions of sets
- Functions, surjections, and injections
- Cardinality of sets
Coordinator Dr. Anthony van Groningen
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