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)

It's really sort of amazing that people manage to communicate in the English language Here are some typical sentences: 1. You may have cake or you may have ice cream 2. If pigs can fly, then you can understand the Chernoff bound 3. If you can solve any problem we come up with then you get an a for the course. 4. Every American has a dream What precisely do these sentences mean? Can you have both cake and ice cream or must you choose just one desert? If the second sentence is true, then is the Chernoff bound incomprehensible? If you can solve some problems we come up with but not all, then do you get an a for the course? And can you still get an a even if you cant solve any of the problems? Does the last sentence imply that all Americans have the same dream or might

1 Strong Induction Recall the principle of strong induction: Principle of Strong Induction. Let(n) be a predicate. If ·P() is true,and for all n, P(O)A P(1)...A P(n) implies P(n+1), then P() is true for all n E N. As an example, let's derive the fundamental theorem of arithmetic

1 Induction Recall the principle of induction: Principle of Induction. Let P(n) be a predicate. If ·P(0) is true,an for all nE N, P(n) implies P(n+1), then P(n) is true for all nE N As an example let's try to find a simple expression equal to the following sum and then use induction to prove our guess correct 1·2+2·3+3:4+…+n·(mn+1) To help find an equivalent expression, we could try evaluating the sum for some small n and(with the help of a computer) some larger n sum

Here we'll discuss the basic idea behind an engine, and then go into detail about how all the pieces fit together and how to increase performance Internal Combustion Almost all cars currently use what is called a four-stroke combustion cycle to convert gasoline into motion. The four-stroke approach is also known as the otto cycle

Defining Interest Groups and Political Parties What is an Interest Group? an organized group of individuals who share common goals or objectives who attempt to influence policymakers in all three branches of government, and at all levels What is a political party?

Ladies and gentlemen: I have come to a decision which I felt I should tell you and all of my fellow American citizens, as soon as I was certain in my own mind and in my own conscience that it is the right thing to do. I have learned already in this* office that the difficult decisions always come to this desk. I must admit that many of them do not look at all the same as the hypothetical questions that I have answered f reely and perhaps too f ast on previous occasions

Vice President Johnson, Mr. Speaker, Mr. Chief Justice, President Eisenhower, Vice President Nixon, President Truman, Reverend Clergy, fellow citizens: We observe today not a victory of party, but a celebration of freedom -- symbolizing an end, as well as a beginning -- signifying renewal, as well as change. For I have sworn before you and Almighty God the same solemn oath our forebears prescribed nearly a century and three-quarters ago. The world is very dif ferent now. For man holds in his mortal hands the power to abolish all forms of human poverty and all forms of human life. And yet the same

Tonight I want to talk to you on a subject of deep concern to all Americans and to many people in all parts of the world, the war in Vietnam. I believe that one of the reasons for the deep division about Vietnam is that many Americans have lost confidence in what their Government has told them about our policy. The American people cannot and should not be asked to support a policy which involves the overriding

...And during the few moments that we have lef t, we want to have just an off-the-cuff chat between you and me -- us. We want to talk right down to earth in a language that everybody here can easily understand. We all agree tonight, all of the speakers have agreed, that America has a very serious problem. Not only does America have a very serious problem, but our people have a very serious problem