Please print out this problem set and record your answers on the printed copy. Answers to this problem set are to be turned in at the box by 4: 10 Wednesday, September10. Problem sets will not be accepted late. Solutions will be posted on the web

Problem Set 10 Solutions Due: Monday, May 2 at 9 PM Problem 1. Justify your answers to the following questions about independence. (a)Suppose that you roll a fair die that has six sides, numbered 1, 2, ... 6. Is the event that the number on top is a multiple of independent of the event that the number on top is a multiple of 3?

Srini Devadas and Eric Lehman Problem Set 7 Solutions Due: Monday, April 4 at 9 PM Problem 1. Every function has some subset of these properties: injective

Problem Set 4 Solutions Due: Monday, February 28 at 9 PM Problem 1. Prove all of the following statements except for the two that are false; for those, provide counterexamples. Assumen 1. When proving each statement, you may assume all its predecessors (a)a =(mod n) Solution. Every number divides zero, so n (a-a), which means a a (mod n). (b)a≡b(modn) impliesa(modn)

Problem set 2 Solutions Due: Monday, February 14 at 9 PM Problem 1. Use induction to prove that n/n for alln olution. The proof is by induction on n. Let P(n) be the proposition that the equation Base case. P(2 )is true because Inductive step. Assume P(n)is true. Then we can prove P(n +1)is also true as follows

Problem set 1 Solutions Due: Monday February 7 at 9 PM Problem 1. The connectives A(and), V(or), and =(implies)come often not only in com uter programs, but also everyday speech. But devices that compute the nand operation

Random walks 1 Random walks a drunkard stumbles out of a bar staggers one step to the right, with a canal lies y steps to his right. Thi I equal p second, he either staggers one step to the left or probability. His home lies r steps to his left, and everal natural questions, including 1. What is the probability that the drunkard arrives safely at home instead of falling into the canal? 2. What is the expected duration of his journey however it ends? The drunkard's meandering path is called a random walk. Random walks are an im- portant subject, because they can model such a wide array of phenomena. For example

The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable r defined on a sample space s is: (B)=∑R()Pr(o) To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let r be the selected student's exam score. Then

Electric Fields 918.1 Conductors in electric field 1. Conductors A conductor is a material in which the electrons at the outer periphery of an atom have no great affinity for any particular individual

Solubility limit for P at 1273 K is Co 2 1 x 10 cm\. Intrinsic carrier concentration is 10 cm\. Thus, before including higher order terms, Do=13x10-4 cm2/s. But with first order and second order correction terms