Homework 3 Solutions

Homework 3 Solutions - EECS 203 Homework 3 Solutions...

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Unformatted text preview: EECS 203 Homework 3 Solutions Section 1.6 6. and (an odd number) 24. Proof by contradiction: Assume that at most two days are selected from any month and still select as least 25 days. To select as many days as possible, we can select two from each month. This yields 212 = 24 days. 2425. This contradiction implies that the assumption (of two days per month) is incorrect. 40. Considering the sums of each of the ten groups of three in this circle, the sums of all such groups must be 165=3(1+2+...+10), because every number from 1 to 10 is in exactly three groups. The average sum of a group is 165/10 = 16.5 From exercise 39, we have that one of these ten sums must be greater than 16.5. The smallest whole number greater than 16.5 is 17, so some group must have a sum 17. (Note: it is also possible to prove this fact indirectly) Section 1.7 14. This can be proved either constructively: , which is a formula to uniquely determine x for each (a,b,c). or by contradiction: Assume there are two distinct values of x that satisfy As above, and . : and with . Thus , which is a contradiction, and means that the assumption of multiple distinct values for x is incorrect. 24. Assume, for the sake of contradiction, that some iteration contained nine zeros. Then in the previous iteration, there were either nine ones or nine zeros. Assume without loss of generality that the previous iteration contained nine ones (since we are already examining a state with nine zeros and the original state contained several ones). If a state had nine ones, then every adjacent pair of bits would have had to be different in the previous iteration. Since there are an odd number of bits in the circle, there must always be at least two adjacent bits that are equal. This contradiction means that the assumption that a state with nine zeros is impossible given the starting state is incorrect. Section 2.1 8. d) true e) true f) true g) false 28.a) b) Section 2.2 14. Students need only produce the correct final sets. The identities are not necessary: ...
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This note was uploaded on 02/12/2012 for the course EECS 203 taught by Professor Yaoyunshi during the Spring '07 term at University of Michigan.

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