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Assign 1 2011 UPDATED

# Assign 1 2011 UPDATED - (copied below 2.3 a Define a...

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Assignment 1: Due Thurs 4/7 Read text, chapters 0, 1, 2 1. What do you know about sets A and B if the following is true? (U = all elements in the universe). a. A B = A b. A B = c. A – B = B d. A B = A B e. A – B = A 2..Show that the following sets are countable (that means construct a one-one and onto function!) a. set of all positive integers divisible by 5 = {5, 10, 15, 20, ….} b. Prove: if sets S and T are countable, then S T is also countable. c, Explain “Hilbert’s Hotel” and what that has to do with countable and/or uncountable. 3. Consider the following languages over Σ = {0, 1}: L 1 = { w : w contains at least two 0’s and | w | 3} L 2 = { w : w contains exactly three 1’s and | w | 4} a. What strings are in L 1 ? b. What strings are in L 2 ? c. What is L 1 L 2 ? d. What is L 1 L 2 ? e. What is L 2 – L 1 ? 4. Consider the string w = abcc. Describe a.. Prefix (w) b. Suffix (w) c. Substring (w) Text, chapter 2 2.3, 2.4, 2.5 a,b, f
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Unformatted text preview: (copied below) 2.3 a. Define a language L smaller-than-w If w = aabbc, under your definition, what is L smaller-than-w ? b. For strings r,s,v,w with r < s and v < w, prove or disprove (i) rv < sw (ii) v R < w R ( iii) w R < v R 2.4. Suppose Σ = {a,b,c} and v = v 1 ,..., v n , n = 3, v = abbc. Give all possible divisions of v into three substrings, v = v 1 v 2 v 3 . 2.5. Let Σ = {a,b,c} with languages L ε = {ε}, L ∅ = , ∅ L 3k = { w so that |w| = 3k for some k >= 0 }, L a = { w | w contains an a} Find: a. L ∅ L 3k b. L ε L 3k f Suffix (L a ) 5. a. If w = aac and x = cb, list all the strings in Shuffle(w,x) and in Shuffle(x,w) b. If L 1 = {a, aa, aab, abc, cba} and L 2 = {a, ba}, list the strings in L 1 / L 2 c. If L 1 = {strings over a,b,c with even number of a’s} and L 2 = {strings with at least one a over a,b,c}, describe the strings in L 1 / L 2...
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