Discrete Mathematics Demystified

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Exercises

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1 Prove that the product of two odd natural numbers must be odd 2 Prove that if n is an even natural number and if m is any natural number then n m must be even 3 Prove that the sum of the squares of the rst n natural numbers is equal to 2n 3 + 3n 2 + n 6 4 Prove that if m is a power of 3 and n is a power of 3 then m + n is never a power of 3 5 Prove that if n is a natural number and if n has a rational square root then in fact the square root of n is an integer 6 Prove that if the natural number n is a perfect square then n + 1 will never be a perfect square 7 A popular recreational puzzle hypothesizes that you have nine pearls that are identical in appearance Eight of these pearls have the same weight, but the ninth is either heavier or lighter you do not know which You have a balance scale, and are allowed three weighings to nd the odd pearl How do you proceed Now here is a bogus proof by induction that you can solve the problem in the rst paragraph in three weighings not just for nine pearls but for any number of pearls For convenience let us begin the induction with the case n = 9 pearls By the result of the rst paragraph, we can handle that case Now, inductively, suppose that we have an algorithm for handling n pearls We use this hypothesis to treat (n + 1) pearls From the (n + 1) pearls, remove one and put it in your pocket There remain n pearls Apply the npearl algorithm to these remaining pearls If you nd the odd pearl then you are done If you do not nd the odd pearl, then it is the one in your pocket That completes the case (n + 1) and the proof What is the aw in this reasoning [Note: If you are endishly clever, then you can actually handle 12 pearls in the original problem with just three weighings However, this requires the consideration of 27 cases] 8 Prove that if k is a natural number that is greater than 2 then 2k > 1 + 2k 9 Prove the pigeonhole principle by induction 10 You write 27 letters to 27 different people Then you address the 27 envelopes You close your eyes and stuff one letter into each envelope What is the probability that just one letter is in the wrong envelope

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Set Theory

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31 Rudiments

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Even the most elementary considerations in logic may lead to conundrums Suppose that we wish to de ne the notion of line We might say that it is the shortest path between two points This is not completely satisfactory because we have not yet de ned path or point And when we say the shortest path do we mean that there is just one unique shortest path And why does it exist Every new de nition is, perforce, formulated in terms of other ideas And those ideas in terms of other ones Where does the regression cease The accepted method for dealing with this problem is to begin with certain terms (as few as possible) that are agreed to be unde nable These terms should be so simple that there can be little argument as to their meaning But it is agreed in advance that these unde nable terms simply cannot be de ned in terms of ideas that have been previously de ned Our unde ned terms are our starting place In modern mathematics it is customary to use set and element of as unde nables A set is declared to be a collection of objects (Please do not ask what an object is or what a collection is; when we say that the term set is an

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