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Oct 18, 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: None) 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 Anthony van Groningen
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