CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Counting Cartesian Products For two sets A and B, define the cartesian product as A×B = {(a,b) : a ∈ A,b ∈ B}. (a) Given two countable sets A and B, prove that A × B is countable. (b) Given a finite number of countable […]
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 Due: Friday 9/3, 10:00 PM Grace period until Friday 9/3, 11:59 PM Before you start writing your final homework submission, state briefly how you worked on it. Who else did you work with? List names and email addresses. (In case of homework party, you can just
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Prove or Disprove Prove or disprove each of the following statements. For each proof, state which of the proof types (as discussed in Note 2) you used. (a) For all natural numbers n, if n is odd then n2 + 3n is even. (b) Forallrealnumbersa,b,ifa+b≥20thena≥17orb≥3.
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Propositional Logic In order to be fluent in working with mathematical statements, you need to understand the basic framework of the language of mathematics. This first lecture, we will start by learning about what logical forms math- ematical theorems may take, and how to manipulate
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 Due: Saturday 10/02, 4:00 PM Grace period until Saturday 10/02, 5:59 PM Before you start writing your final homework submission, state briefly how you worked on it. Who else did you work with? List names and email addresses. (In case of homework party, you can just
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 In science, evidence is accumulated through experiments to assert the validity of a statement. Mathematics, in contrast, aims for a more absolute level of certainty. A mathematical proof provides a means for guar- anteeing that a statement is true. Proofs are very powerful and are in
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a smaller range of numbers. Modular arithmetic is useful in these settings, since it limits numbers to a prede- fined range {0, 1, . . . , N
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 RSA Warm-Up Consider an RSA scheme with modulus N = pq, where p and q are distinct prime numbers larger than 3. (a) What is wrong with using the exponent e = 2 in an RSA public key? (b) Recall that e must be relatively
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Stable Matching Consider the set of jobs J = {1, 2, 3} and the set of candidates C = {A, B, C} with the following preferences. Jobs Candidates 1 A>B>C 2 B>A>C 3 A>B>C Candidates Jobs A 2>1>3 B 1>3>2 C 1>2>3 Run the traditional
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 The next major topic of the course is probability theory. Suppose you toss a fair coin a thousand times. How likely is it that you get exactly 500 heads? And what about 1000 heads? It turns out that the chances of 500 heads are roughly 2.5%,
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