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Unit of study_
MATH3066: Algebra and Logic
This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times. The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem. Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.
| Code | MATH3066 |
|---|---|
| Academic unit | Mathematics and Statistics Academic Operations |
| Credit points | 6 |
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Prerequisites:
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6 credit points of MATH2XXX |
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Corequisites:
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None |
| Prohibitions:
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MATH3062 or MATH3065 |
| Assumed knowledge:
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Introductory knowledge of group theory. For example as in MATH2X22 |
At the completion of this unit, you should be able to:
- LO1. be fluent in analysing and constructing logical arguments
- LO2. be conversant with the Propositional and Predicate Calculi, and related notions of syntax (deduction) and semantics (completeness)
- LO3. be informed about the historical underpinnings of abstract algebra that lead to axiomatic theories of groups, rings, integral domains and fields, and their use in exploring questions of decidability or impossibility
- LO4. be fluent with a range of standard and exotic arithmetics and ring constructions, and be able to prove elementary propositions and theorems about them
- LO5. understand the Halting Problem and Turing’s use of it to prove the undecidability of first order logic.
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