Notes for recitation 5 1 Well-ordering principle Every non-empty set of natural numbers has a minimum element Do you believe this statement? Seems obvious, right? Well, it is. But dont fail to realize how tight it is. Crucially, it talks about a non-empty set -otherwise, it would clearly be false. And it also talks about natural

1 Independent Events Suppose that we flip two fair coins simultaneously on opposite sides of room. Intu- itively, the way one coin lands does not affect the way the other coin lands. The mathe- matical concept that captures this intuition is called independence. In particular, events A and B are independent if and only if:

Conditional Probability Suppose that we pick a random person in the world. Everyone has an equal chance of being selected. Let A be the event that the person is an MIT student, and let B be the event that the person lives in Cambridge. What are the probabilities of these events? Intuitively

Counting III Today we'll briefly review some facts you dervied in recitation on Friday and then turn to some applications of counting. 1 The Bookkeeper Rule In recitation you learned that the number of ways to rearrange the letters in the wore BOOKKEEPER is:

Sums, Approximations, and Asymptotics II Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table's edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+1)-st block;

Sums and Approximations When you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a load-balancing or communications scheme, you'll rarely get a simple answer. The world is not so kind. More likely, you'll end up with a complicated sum:

1 Introduction normally, a graph is a bunch of dots connected by lines. Here is an example of a graph

Srini devadas and Eric Lehman Lecture notes Number theory ll Image of Alan Turing removed for copyright reasons s The man pictured above is Alan Turing, the most important figure in the history of mputer science. For decades, his

Srini Devadas and Eric Lehman Lecture Notes Induction III 1 Two Puzzles Here are two challenging puzzles. 1.1 The 9-Number Puzzle

Why do you believe that 3+3=6? Is it because your second-grade teacher, Miss Dalrymple, told you so? She might have been lying, you know Or are you trusting life experience? If you have three coconuts and someone gives you three more coconuts, then you have--ahal--six coconuts. But if that is the true basis for your belief, then why do you also believe that