Lecture_Notes (reformatted)

Lecture_Notes (reformatted) - ArsDigita University Month 2:...

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ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Lecture Notes What is Discrete Math? Example of continuous math – Given a fixed surface area, what are the dimensions of a cylinder that maximizes volume? Example of Discrete Math – Given a fixed set of characters, and a length, how many different passwords can you construct? How many edges in graph with n vertices? How many ways to choose a team of two people from a group of n? How many different binary trees (is it worth checking them all to find a minimum spanning tree of a graph – a tree that includes all the vertices of a weighted edge graph, with minimum sum of weights)? How many ways to arrange n arrays for multiplication? How many ways to draw n pairs of balanced parens? Note that the last 3 examples have the same answers (not obvious). Note the second and third examples have the same answer (obvious). Counting is an important tool in discrete math as we will see later. What are proofs? Formal definitions and logic versus… A proof is a clear explanation, accepted by the mathematical community, of why something is true. Examples…. Ancient Babylonian and Egyptian mathematics had no proofs, just examples and methods. Proofs in the way we use them today began with the Greeks and Euclid. 1. The square root of two is irrational - A proof by contradiction from Aristotle. Assume that ± ² √³ , where a and b are relatively prime. Squaring both sides of the equation gives ´µ ¶µ = 2. Then a 2 = 2b 2 , and since an even number is any number that can be written as 2k, a 2 must be even. By a separate lemma, we know that if a 2 is even, then a must also be even. So write a = 2m. Then a 2 = (2m) 2 and a 2 = 4m 2 , and 2b 2 = 4m 2 , so b 2 is even, and b is even. But we assumed without any loss of generality that a and b were relatively prime, and now we have deduced that both are even! This is a contradiction, hence our assumption that ± ² √³ , cannot be right.
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2. There are an infinite number of prime numbers – A proof by contradiction by Euclid. Assume that there is a finite number of prime numbers. Construct their product and add one. None of the prime numbers divide this new number evenly, because they will all leave a remainder of one. Hence, the number is either prime itself, or it is divisible by another prime not on the original list. Either way we get a prime number not in the original list. This is a contradiction to the assumption that there is a finite number of prime numbers. Hence our assumption cannot be correct. Discovering theorems is as important as proving them. Examples: 1. How many pairs of people are possible given a group of n people? Constructive counting method: The first person can pair up with n-1 people. The next person can pair up with n-2 people etc, giving (n-1) + (n-2) + … + 2 + 1 Counting argument: Each person of n people can pair up with n-1 other people, but counting pairs this way, counts each pair twice, once from each end.
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Lecture_Notes (reformatted) - ArsDigita University Month 2:...

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