UCSB CMPSC40 — Foundations of Computer Science — Winter 2021
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Course OverviewIntroduction to the theoretical underpinnings of computer science.
CS 40 introduces the essential mathematical background necessary for computer science, from logical reasoning to a number of basic constructs of discrete mathematics required to succeed in more advanced courses in the curriculum. It is a 5 unit course instead of the usual 4, and as such requires more study time and commitment.
Prereqs: 16 and Math 4A
Prereq for: 130A, 138, 178
Books: “Discrete Mathematics and Its Applications” by Kenneth H. Rosen, 6th edition or newer.
Topics covered in CS40
- Propositional and first-order predicate classical logic
- Proof techniques
- Mathematical datatypes: Sets, function, relations
- Set cardinality and countability
- Induction and recursion, strong induction, structural induction
- Recursively defined structures
- Divide-and-conquer type recurrences
- Combinatorics: Elementary counting
- Combinatorics: Permutations, combinations, binomial coefficients
- Modular arithmetic and elementary cryptography
- The principle of inclusion-exclusion
Lectures and Reading Assignments
|Week 1||Chapters 1.1-1.3 (propositional logic)|
|Week 2||Chapters 1.4-1.8 (predicate logic and proof methods)|
|Week 3||Chapters 2.1-2.3 (sets and functions)|
|Week 4||Chapters 2.3-2.4 (functions, sequences and summation), additional notes on sequences and summation; chapters 4.1-4.3 (number theory)|
|Week 5||Lecture notes (Representation of negative integers, Fermat’s little theorem and RSA encryption); chapters 4.4-4.5 (solving congruences and applications)|
|Week 6||Lecture notes (cardinality and countability)|
|Week 7||Lecture notes (computability and density); chapters 9.1, 9.5, 9.6 (partial orderings, well orderings and equivalence relations)|
|Week 8||Chapters 5.1-5.3 (mathematical induction, strong induction, structural induction and recursive structures)|
|Week 9 & 10||Lecture notes (combinatorics); chapters 6.1-6.5 (combinatorics: counting, combinations and permutations, binomial coefficients and binomial theorem, generalized combinations and permutations)|