Honors Discrete Mathematics
Honors Discrete Mathematics MAD 2104
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This 4 page Class Notes was uploaded by Chaya Botsford II on Monday October 12, 2015. The Class Notes belongs to MAD 2104 at Florida Atlantic University taught by Jorge Viola-Prioli in Fall. Since its upload, it has received 19 views. For similar materials see /class/221637/mad-2104-florida-atlantic-university in Mathematics Discrete at Florida Atlantic University.
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Date Created: 10/12/15
5COUNTEREXAMPLE How do we show that a statement is false In order to answer this question we will introduce the quotevil character mentioned in section 2 the counterexample Assume we need to show that a statement of the form quotlfA then Bquot is false A quick look at the truth table of A gt B shows that the only way for this implication to fail is that A be true but B be false In other words we need to nd one instance in which A is true and B is false Equivalently we must present a particular example called counterexample which makes the statement false Consider for instance the following assertion quotIf n is a positive integer then Pn n2 n 11 is a prime number Let us see what happens when we evaluate P at different values of n J Viola Prioli P1 13 P2 17 P3 23 P4 31 P5 41 P6 53 P7 67 P8 83 P9 101 Observe that each ofthe values obtained is a prime number We have not proved the statement yet however because we have not evaluated P at every n as required in the assertion Next we plug n 10 and get P10 121 11 x 11 so P10 is not a prime We have produced a counterexample thus showing that the statement is false To disprove a statement is an alternative way of asking to provide a counterexample A famous conjecture by Fermat claims that for every natural n the number Pn 22 1 is prime The computations show that PO 3 P1 5 P2 17 P3 257 and P4 65537 each of these resulting numbers is prime But is the conjecture true J Viola Prioli Approximately one hundred years after Fermat39s claim Euler 1739 produced a counterexample actually he showed that P5 that is 232 1 equals 641 x 6700417 and so P5 is not a prime number Another instance which we may encounter is to disprove a statement of the form A ltgt B What does a counterexample need to satisfy in such an instance A look at the corresponding truth table shows that A ltgt B is false only when A is true and B is false or when A is false and B is true In other words a counterexample must show that one of the implications fails perhaps both fail but showing that one fails suf ces J Viola Prioli For instance disprove that quota positive integer n is composite if and only if it has two different prime divisors We notice that this is A ltgt B where A a positive number n is composite and B n has two different prime divisors In our case it is clear that B gt A is true However A gt B fails since 9 is composite so A is true but 3 is its only prime divisor so B is false Summing up n9 is a counterexample to the whole statement quota positive integer n is composite if and only if it has two different prime divisors As a consequence the statement is false J Viola Prioli
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