hmwk_2

# hmwk_2 - • the set of all integers • all ﬁnite...

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Discrete Structures Aug 30 2011 Assignment 2: Due at beginning of class Mon, Sept 12 Prof. Hopcroft *******Note: this is the ﬁrst homework that will be subject to the style guide on the website. Points will be deducted for homeworks not following the guidelines.******** 1 Regular Expressions Write a regular expression for the set of all strings of 0’s and 1’s that have two blocks of 1’s. A sample string your regular expression should represent: 000000111111000111000 2 Recurrence Relation Find a close form solution for the following recurrence equations: (Also, show that you checked the ﬁrst four values of your solution) 1. f ( n ) = 5 f ( n - 1) - 6 f ( n - 2) with boundary conditions f (0) = 0 and f (1) = 1. 2. f ( n ) = 6 f ( n - 1) - 12 f ( n - 2)+8 f ( n - 3) with boundary conditions f (0) = 1, f (1) = 6, and f (2) = 32. 3 Induction Prove by induction that 2 n 4 + 2 n 2 + 4 is divisible by 4 for all n 1. 4 Countably Inﬁnite OR NOT!?! For each set say whether or not the set is countably inﬁnite:

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Unformatted text preview: • the set of all integers • all ﬁnite subsets of integers • all subsets of integers • the set of all reals • all ﬁnite subsets of reals • all subsets of reals • the set of all ﬁnite length strings • all ﬁnite subsets of the set of all ﬁnite length strings • all subsets of the set of all ﬁnite length strings • all computer programs 1 • functions mapping integers to the set { , 1 } • fuctions mapping integers to integers 5 Countable Concept In a concise way, write an explaination of countably inﬁnite a high school student could understand. And, give an example of a set that is not countably inﬁnite or ﬁnite and explain why. 6 Countable Triplets Sketch a algorithm to list all triples of a countably inﬁnite set. 2...
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hmwk_2 - • the set of all integers • all ﬁnite...

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