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# hw4cs - 1 Is this a directed graph 2 Is this graph...

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CSE 20: Discrete Mathematics Fall 2010 Problem Set 4 Instructor: Daniele Micciancio Due on: Wed. Nov 3, 2010 Problem 1 (6 points) Let A, B and C be subsets of U . Prove or disprove each of the following statements using Venn diagrams: 1. ( A \ B ) \ C = A \ ( B C ) 2. ( A \ C ) ( B \ C ) and ( A \ B ) are disjoint 3. ( A B ) C = A ( B C ) Problem 2 (5 points) Let A = { 1 , 3 } , B = { 5 , 6 } and C = { 1 , 3 , 5 , 7 , 9 } . List the elements for each of the following sets: 1. ( A B ) C 2. ( A × B ) A 3. ∅ × ∅ 4. { x C | ( x + 1) A B } 5. ( C \ ( A B )) 2 Problem 3 (8 points) Let A and B be subsets of a universal set U . Use the definition of the set operations (as given in class an in section 2.2 of the textbook) and the rules of logic to prove the following identity: ( A B ) ( A 0 B 0 ) = ( A B 0 ) ( A B 0 ) 0 Problem 4 (6 points) [Problem 2.1(9) from the textbook] Think of the internet as one big graph, where each page is a vertex and each link is an edge. Answer the following questions, and provide a brief (one line) explatation:

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Unformatted text preview: 1. Is this a directed graph? 2. Is this graph connected? 3. Is this graph complete? 4. In this graph simple? 5. For a given webpage p, what does the outdegree of p represent? 6. For a given webpage p, what does the indegree of p represent? Problem 5 (6 points) [Problem 2.1(13) from the textbook] Using as few groups as possible, put the words “ﬁsh”, “sit”, “stay”, “play”, “diet”, “tree”, “duck”, “dog”, and “hen” into groups such that none of the words in a group have any letters in common. Use a graph model and graph coloring. Justify your answer: explain why your grouping uses the fewest groups as possible....
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