propositional_logic_and_quantifiers - CS 70 Fall 2006...

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CS 70 Discrete Mathematics for CS Fall 2006 Lecture 1 Course Outline CS70 is a course about on ”Discrete Mathematics for Computer Scientists”. The purpose of the course is to teach you about: Fundamental ideas in computer science: - Boolean logic. - Uncomputability and the halting problem. - Modular arithmetic, Error-correcting codes, secret sharing protocols - Graphs: paths, cuts, hypercubes. Many of these concepts underly all the more advanced courses in computer science. Precise, reliable, powerful thinking: - Proofs of correctness. These are essential to analyzing algorithms and programs. - Induction and recursion. - Probability theory. Problem solving skills: - These are emphasized in the discussion sections and homeworks. Course outline (abbreviated). • Propositions, Propositional logic and Proofs • Mathematical Induction, recursion • The stable marriage problem • Modular arithmetic, the RSA cryptosystem • Polynomials over finite fields and their Applications: error-correcting codes, secret sharing • Graphs: Eulerian paths, hypercubes. • Diagonalization, Self-Reference, and Uncomputability • Probability and Probabilistic Algorithms: load balancing, hashing, expectation, variance, Chebyshev and Chernoff bounds, conditional probability, Bayesian inference, law of large numbers, power laws. CS 70, Fall 2006, Lecture 1 1
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Lesson Plan In order to be fluent in mathematical statements, you need to understand the basic framework of the language of mathematics. This first week, we will start by learning about what logical forms mathematical theorems may take, and how to manipulate those forms to make them easier to prove. In the next few lectures, we will learn several different methods of proving things. Propositions A proposition is a statement which is either true or false. These statements are all propositions: (1) 3 is irrational. (2) 1 + 1 = 5. (3) Julius Caesar had 2 eggs for breakfast on his 10 th birthday. These statements are clearly not propositions: (4) 2 + 2 (5) x 2 + 3 x = 5. These statements aren’t propositions either (although some books say they are). Propositions should not include fuzzy terms. (6) Schwarzenegger often eats broccoli. (What is “often?”) (7) George W. Bush is popular. (What is “popular?”) Propositions may be joined together to form more complex statements. Let P , Q , and R be variables repre- senting propositions (for example, P could stand for “3 is odd”). The simplest propositional forms combine variables using the connectives “and, or, and not.” (1) Conjunction : P Q (“ P and Q ”). True only when both P and Q are true. (2) Disjunction : P Q (“ P or Q ”). True when at least one of P and Q is true.
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propositional_logic_and_quantifiers - CS 70 Fall 2006...

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